Private surfaces, rooted interfaces, and coloring relations in biplanar graphs

1. Statement and earlier work

A graph is biplanar if its edges partition into two planar graphs on the same vertex set. The embeddings of the layers may differ. We call an operation ambient when all edges outside the deleted private set survive, subject only to specified identifications of independent terminals.

This revision adds four ambient reduction results. Theorem 23 compresses every possible common neighborhood of a private six-clique to its actual chromatic number, without a connectivity assumption on the private layers. Theorem 24 finds a nonadjacent pair of any five common neighbors of a five-clique that can be identified through the private vertices in both layers; no terminal-edge restriction is imposed. Theorem 25 compresses six common terminals when two are exclusive to opposite layers; an actual one-exclusive drawing shows why that routing hypothesis matters. Theorem 26 treats a five-clique complete to the eight-vertex Wagner complement: every representation admits four simultaneous independent-pair identifications. These operations preserve the arbitrary exterior and exclude the specified true-twin configurations from a globally minimum-order counterexample. The six-clique argument imports a published computational obstruction. The new five-terminal normalization uses a checked Kuratowski family and an earlier common-face obstruction; the eight-terminal representation classification uses a separate LRAT premise. None forces these cliques to occur in an arbitrary counterexample.

Theorem 1 (retained small-terminal theorem). Suppose $G$ is biplanar, $Q$ is a clique of order six, and every vertex of $Q$ has the same outside neighborhood $S$, with $|S|\leq6$. There is a smaller biplanar graph such that every nine-coloring of it extends to a nine-coloring of G. Thus a minimum-order biplanar graph not nine-colorable contains no such configuration.

For six terminals the hard-case quotient has nine fewer vertices; for the five-cycle boundary it has eight fewer. The reduction is made in each actual layer, not only in a specially chosen drawing of the local graph.

The accepted mandatory four-terminal attachment theorem [1] already gives a private connected component reaching any four prescribed common terminals. It supplies the basic single-group identifications. We credit that theorem rather than claim attachment or those operations as new. Prior smoothing work [2] coordinates two disjoint pairs in particular interfaces, while earlier common-face arguments [3] use drawable stars and chords. The ingredients of the original reduction are the contact-tree description, two-group routing through a private surface, and the treatment of small secondary components. An earlier five-terminal majority argument was shared in [4]; the present proof gives a stronger and shorter treatment of that boundary and extends to six terminals.

We use the classical fact that $K_{9}$ is not biplanar [5]. All graphs are finite and simple; intermediate contractions may create parallel edges, which are suppressed at the end. The word "face" always refers to the fixed embedding under discussion.

A second operation deletes a whole induced $K_6\vee C$ core. With at most one outside neighbor at every clique vertex, rooted component stars ensure that any common clique-forbidden color is forbidden only on an independent set of cycle vertices. This gives reductions at total cycle outside-degree excess at most seven, and, separately for $C_5$, when every cycle vertex has at most four outside neighbors. A partial-cone argument also handles one exceptional clique vertex with up to eight outside neighbors when all other core vertices have at most one. A selected-spoke lemma then handles mixed clique/cycle exceptions without the finite attachment premise. Allowing one or two fresh quotient vertices passes sharp barriers to edge-only universal extension; a bilateral boundary incidence condition supplies two globally available cycle colors. Section 13 proves these statements and distinguishes local auxiliary mergers from fixed ambient quotients. Section 14 gives a finite face-system obstruction to a natural 17-vertex equality carrier, with a compact RUP certificate.

Szeider's global independence-number-two theorem [6] establishes nine-colorability under a restriction on the entire graph. Our hypotheses concern a specified local core and allow an arbitrary ambient exterior; that global theorem does not subsume these reductions.

2. A dominant private component

Lemma 2. In either layer of a planar partition of $K_{6}$ joined to an independent set S with $|S|\geq4$, some private component C has at least four vertices and reaches every terminal. At most two private vertices remain outside C.

Proof. If the private graph is connected, the known four-terminal attachment theorem [1], applied to subsets containing each terminal, ensures that every terminal has a spoke to this component.

If it is disconnected, a union of private components of order three would put all cross edges of a 3+3 split in the opposite layer, giving $K_{3,3}$. The remaining component-size partitions are $5+1$, $4+2$, $4+1+1$ and $2+2+2$.

For $2+2+2$, the opposite private graph is the octahedron $K_{2,2,2}$, a maximal planar graph. A terminal in that layer can meet at most one member of each original pair, because it lies in a triangular face. In the original layer every terminal therefore meets all three pair components. Contracting them gives $K_{3,|S|}$, a contradiction.

For a component of order four, a terminal missing it would join both remaining private vertices as three opposite-layer vertices complete to that four-set, giving $K_{3,4}$.

For the $5+1$ split, let z be the singleton and suppose s misses the large component. In the opposite layer s and z are complete to its five vertices, forming $K_{2,5}$. Its embedding gives a cyclic order B of those five vertices. Every opposite-layer private edge joins consecutive B vertices, and every other terminal's private neighbors there lie in a consecutive B-pair, since it occupies a quadrilateral face of $K_{2,5}$.

The original layer therefore contains the complementary pentagon $D=K_5-E(B)$. Each other terminal has neighbors containing the complement of a consecutive B-pair, a triple with cyclic gaps 1,2,2 on D. At most one such star fits on either side of D: the first star divides that side into sectors, each meeting at most three consecutive D vertices, and none of those sectors contains another triple of gaps 1,2,2. There are at least three other terminals, so two sides cannot accommodate them. Hence C reaches all terminals. QED.

The complementary-pentagon argument is the same geometry used in the private-seven-clique work with lunar_structure [2]. The current result uses this as structural support, not as its novel conclusion.

3. Cofacial completion

In each layer restrict the actual drawing to $Q\cup S$ and remove all terminal-terminal edges. No resulting face can touch three distinct terminals: add their triangle in that face and retain the clique and every private-terminal join across the two layers, obtaining $K_{9}$. A face with repeated boundary vertices or holes still permits that triangle: join three chosen terminal corners by an interior tree and trace a thin neighborhood of its three branches.

Add each pair of cofacial terminals in one supporting face of its layer. These additions can be performed simultaneously, since no face supports more than one distinct terminal pair. Let H be their union across the two layers. Then H is triangle-free, by the same $K_{9}$ obstruction. Moreover, H contains $G[S]$: deleting an original terminal edge leaves its endpoints cofacial in the layer that held it.

This is an auxiliary local completion. The added H edges need not coexist with the arbitrary ambient exterior. They will not be used as ambient edges; only proper partitions and degree bounds for H will be used.

For reference, $e(H)\leq9$ for any size of S: the local graph $K_{6}$ joined to H has 6+|S| vertices and 15+6|S|+$e(H)$ edges, and the two planar-layer Euler bound gives $15+6|S|+e(H)\leq6(6+|S|)-12$. This bound is not required for the six-terminal theorem.

4. Contact trees in a connected private layer

Suppose the private graph C in one layer is connected and reaches every terminal. Take a sufficiently thin regular neighborhood $N(C)$. It is a connected planar surface, with one boundary circle for every private face. Its complementary components on the sphere are disks. Each terminal lies in one such disk, and its C-spokes form a star whose windows are distinct points of that disk's boundary.

For the moment suppose every region between these terminal stars touches at most two distinct terminals, as it does for a connected private layer in Section 3. Form a bipartite incidence graph of terminal-star centers and the regions between them, with incidences for occurrences of centers on region boundaries. It is a tree. Insert the stars in any order. A star of d spokes lies in one previous region and replaces that region by d regions joined through the new star node. Old center incidences are distributed among the new regions. This preserves the tree property. Distinct sectors of one star stay in distinct regions, so there are no repeated incidences between a star and a region.

Every region node has degree one or two. Delete degree-one region nodes and suppress degree-two region nodes. The resulting plane tree T has exactly the terminals of that private face as vertices. Its edges are cofacial terminal pairs, so T is a subgraph of H. Different private faces give a plane forest. All rotations are those of the actual drawing.

The contour word of T equals the cyclic attachment-window word after consecutive equal labels are consolidated. To see this precisely, keep the full incidence tree. Its contour with region labels omitted is the window word: the d sectors at a star are separated by its d spokes, so contour corners correspond to successive windows. The star-insertion construction also proves this identity inductively. Deleting a leaf region removes a detour s-region-s, hence one consecutive repeat of s. Suppressing a degree-two region removes only a label already omitted. A single-terminal tree has the constant one-label word. Consolidation does not change the existence of a choice of one window per used terminal.

5. A window lemma and two-tree surface routing

Lemma 3. In a subcubic plane tree properly partitioned into at most three A vertices, at most two B vertices, and an unused singleton x, choose one contour occurrence of each A/B vertex so that the two groups form separate cyclic blocks.

Proof. With at most one B this is immediate. Otherwise let P be the path between the two B vertices. Choose endpoint corners alongside P, keeping endpoint branches outside the selected interval. Every internal A vertex on P has an occurrence on either side. There is at most one internal branch: two would force a four-vertex P and one extra leaf at each internal vertex, using all six vertices. The adjacent internal vertices would be A and x, but the extra leaf at the A vertex could be neither A, B (both already endpoints), nor x (already used). Choose the side without the possible branch and select all A occurrences outside the resulting B interval. QED.

Lemma 4. Let marked points on boundary circles of a connected planar surface be assigned to two groups. If on each boundary circle the groups occupy separate cyclic blocks, disjoint embedded trees join each group, with marked points as leaves.

Proof. On every circle with first-group points choose a boundary interval containing those points and no second-group point. Join the first-group points by a tree in a thin collar of that interval, with marked points as leaves. The small complementary caps have no second-group points. Retain the connected main surface, whose corresponding boundary circle now runs along the inward side of that tree. Different original circles still give different boundary circles. A singleton needs only an attachment point.

Join two first-group trees across their distinct boundary components by an arc in the main surface. Cutting along that arc keeps the main surface connected and merges those two boundary components. Inductively join the accumulated tree to a tree on a still distinct boundary component. Every arc joins distinct tree components, so their union is a tree, and all second-group points stay in a connected main region. Join them there by an embedded tree disjoint from the first. Use attachment sites away from marked leaves. Empty groups need no construction. QED.

Applied to $N(C)$, with extensions along selected old spoke strips, these are ambient routes after the old private material is deleted. Thin neighborhoods avoid every edge not incident with Q. Different routes may use different sectors of the same old private vertex; they are not required to be vertex-disjoint paths of the unsplit private graph. This is the freedom supplied by retaining the whole surface rather than an arbitrary private spanning tree.

Corollary 5. If $|S|=6$, $\alpha(H)\leq3$, and the private layer is connected, every proper $3+2+1$ partition of H is realizable by two disjoint private routing trees in that actual layer.

Indeed H is subcubic: each neighborhood is independent in a triangle-free graph. Its contact trees are subcubic and properly colored by the chosen partition. Apply Lemmas 3 and 4, face by face and then through $N(C)$.

6. Small secondary components

Let C be the dominant private component in a disconnected layer. Every other private component D has order at most two. Put $W=N(D)\cap S$.

Lemma 6. If $|W|\geq3$, the terminal graph H contains a cycle spanning W. If $|W|\leq2$, deleting D preserves terminal cofaciality in this restricted local layer. In particular $|W|=3$ is impossible.

Proof. If D is an edge, contract it. This preserves the terminal incidences on face boundaries. Parallel spokes to a terminal a can arise. Two such spokes bound a Jordan curve; the connected C is on one side. The disk on the other side contains no other terminal, because every other terminal has a spoke to C disjoint from that curve. When D has two vertices all remaining private vertices are in C, so that disk has no other private vertex either. Remove the redundant spoke across this empty disk. Its erased face has only terminal a, so no terminal cofacial pair changes. After all such removals D is a simple star.

If it has at least three leaves, consecutive spokes give face-boundary segments a-d-b. The cyclic order of its distinct terminal leaves therefore supplies a cycle in H. Three leaves would give a triangle.

If it has two leaves it is a path a-d-b. Both adjacent face boundaries contain a,b and no third terminal. Deleting the path merges only those terminal sets, preserving cofaciality. One leaf is a pendant edge and zero leaves an isolated point. The argument for a singleton D also holds when another secondary singleton remains elsewhere in the face. QED.

The contractions in this proof are auxiliary local operations. Actual routes below use the original secondary component and its spokes.

Lemma 7. Let H be triangle-free on six vertices, and suppose W carries a cycle of length $k\in\{4,5,6\}$. Then H has a proper partition $A\mid B\mid\{x\}$ of sizes $3+2+1$ with B contained in W.

Proof. If $k=4$, $H[W]=K_{2,2}$, with opposite pairs P,R. Any outside vertex y misses one entire opposite pair P, because neighbors in both pairs would make a triangle. Set $A=P\cup\{y\},\ B=R$; the other outside vertex is x.

If $k=5$, $H[W]=C_5$. The outside vertex y has at least three nonneighbors on the cycle, containing a nonadjacent pair P. Let $A=P\cup\{y\}$, and choose B as a nonedge among the other three cycle vertices.

If $k=6$, triangle-freeness gives an independent triple A: either a vertex has three neighbors, or all degrees are at most two and the path/cycle components give such a triple. The remaining three vertices contain a nonedge B, automatically inside W. QED.

7. Proof for six terminals

If $\alpha(H)\geq4$, identify an independent four-set through a private component in each layer, using Lemma 2 (or [1] directly for that four-set). Three terminal vertices remain.

Otherwise H is subcubic and has a proper $3+2+1$ partition: choose an independent triple by the elementary triangle-free six-vertex argument in Lemma 7, and then a nonedge among the other three vertices.

At least one private layer is connected, because the complement of a disconnected graph on Q is connected. If both are connected, Corollary 5 realizes any one such partition in both.

Consider the disconnected layer. If some secondary component reaches at least four terminals, Lemmas 6 and 7 give a compatible partition whose pair B is reached by that component. Join B through it and A through C. These trees have disjoint private interiors and disjoint terminal sets. The connected opposite layer supports the same partition by Corollary 5.

Otherwise every secondary component reaches at most two terminals, since three is impossible. Delete them one by one by Lemma 6. Cofaciality is unchanged, so the remaining C plus its terminal stars still has subcubic contact trees contained in H. Corollary 5 applies to this layer also.

In each case delete Q, draw the two routing trees in the freed private material, and contract them. The identified groups are independent in G because they are independent in H. Suppress parallel edges and assign any duplicated original edge to one layer. The quotient is biplanar, retains every exterior edge modulo the declared identifications, and has nine fewer vertices. Its three terminal vertices use at most three colors; lifting its nine-coloring and giving Q six unused colors proves the six-terminal case of Theorem 1.

8. Fewer terminals and stronger five-cycle geometry

With at most three terminals, delete Q. With four, identify any nonadjacent pair through the dominant private components. With five, an independent triple can similarly be identified, leaving three terminal vertices.

The remaining five-terminal case is $H=C_5$. Indeed a triangle-free five-vertex graph with no independent triple has all degrees exactly two: three neighbors would be independent, while a vertex of degree at most one would have three nonneighbors, among which a nonedge gives an independent triple with that vertex. Thus it is a five-cycle.

If the private layer is connected, all contact trees lie in $C_{5}$, hence are paths. In a plane path any two groups of its vertices can have separate contour blocks: choose occurrences of the first group on one side of the path and occurrences of the second on the other. End vertices have one occurrence and can be assigned to the corresponding ends of these blocks. Lemma 4 therefore realizes every two-pair partition of the five terminals.

In a disconnected layer, Lemma 6 leaves only secondary supports of size at most two or size five: a three-cycle or four-cycle cannot lie in $H=C_5$. A five-terminal secondary component routes either chosen pair, while C routes the other. If all supports have size at most two, delete them and apply the preceding path argument.

Thus each actual layer supports all five proper $2+2+1$ partitions of $C_{5}$. In fact its geometry supports all fifteen disjoint two-pair partitions, although adjacent pairs cannot be identified in a proper coloring. Choose one proper partition in both layers, delete Q and contract the two pairs. The quotient has eight fewer vertices and three terminal vertices; six unused colors extend to Q. This proves Theorem 1 in every case.

9. Controlled extra attachments

Corollary 8. Suppose Q is a six-clique complete to a common set S with $|S|\leq6$. Let a(q) be the number of its neighbors outside Q union S, and arrange these counts as $a_1\geq\cdots\geq a_6$. If $a_i\leq6-i\quad(i=1,\ldots,6)$, the configuration is reducible in a minimum-order biplanar graph not nine-colorable.

Delete the extra private incidences before constructing the routes, while retaining their outside endpoints and all exterior edges. The same quotient supplies a coloring of G-Q using at most three colors on S. The available list at $q_i$ has size at least $6-a_i\geq i$. Color Q in the order $q_1,\ldots,q_6$; at step i at most i-1 colors have been used in the clique. This extends the coloring. The common set is essential to this statement; no arbitrary nonuniform degree-ten or degree-eleven clique theorem follows.

10. Optimal compression of a common five-set

Theorem 9 (five-terminal compression). Let a private six-clique Q be complete to a common five-set S in a biplanar graph. After deleting Q and its incidences, an ambient quotient can reduce S to $\chi(G[S])$ vertices. Extra outside neighbors of Q may be present; their incidences are deleted and endpoints retained.

The statement concerns the available number of terminal colors, not a claim that every prescribed bipartition is routable in every completion.

Proof. The local cofacial completion H is triangle-free on five vertices. If $G[S]$ is edgeless, identify all five through the dominant private component in each layer. This is a single-group route.

If H is bipartite, choose a bipartition of H with smaller part of size at most two. The contact-tree two-block proof works for that partition: the path between two vertices of the smaller part has no internal branch, since an off-path neighbor would belong to that exhausted smaller part. The two-tree surface lemma therefore routes its parts in every connected private layer. In a disconnected layer, secondary supports of size at least three must carry a $C_{4}$. Route one opposite pair through that secondary component, and the other opposite pair plus the fifth vertex through the dominant component, choosing the pair missed by the fifth vertex. If all supports have size at most two, delete them locally without changing cofaciality and apply the contact argument. The opposite private layer is connected and supports the same proper bipartition. Thus S is reduced to two vertices whenever it has an edge and H is bipartite.

The only nonbipartite triangle-free H on five vertices is $C_{5}$. If $G[S]$ is bipartite, choose any proper bipartition A|B of $G[S]$; it need not be proper in H. Every contact tree in a connected private layer is a path. For ANY two groups of vertices of a plane path, select A windows on one side and B windows on the other. The contour traverses the path forward then backward. Selected occurrences form two cyclic blocks, including when an endpoint belongs to either group. This argument uses no independence in H. Apply the two-tree surface lemma.

In a disconnected layer with $H=C_5$, a secondary support has size at most two or size five. In the latter case route A through that component and B through the dominant one; arbitrary group sizes cause no difficulty for a single-group tree. Otherwise delete the small secondary components and use the path argument. Thus the same proper bipartition of $G[S]$ works in both layers. Every retained exterior edge survives.

Finally $G[S]$ nonbipartite forces $G[S]=H=C_5$, which the existing two-pair reduction reduces to three vertices. These are exactly its three required colors. QED.

One-extra consequence. If every q has at most one additional neighbor outside Q+S and $G[S]$ is bipartite, two terminal colors leave at least seven palette names, and every private list has size at least six. Greedy coloring extends to $K_{6}$. Therefore a minimum-order non-nine-colorable biplanar graph with this common-five/one-extra configuration must have ACTUAL $G[S]=C_5$, not merely a $C_{5}$ cofacial completion.

More generally, for a bipartite common five-set the sorted extra counts $b_1\geq\cdots\geq b_6$ may satisfy $b_i\leq7-i$. Lists then have size at least i and color in that order. This shifts the earlier staircase bound by one.

11. One extra neighbor and three private ports

Lemma 10 (Hall obstruction). Suppose every private vertex has at most one extra neighbor outside $Q\cup S$. After a quotient colors S with at most three colors, extension to Q fails exactly when S uses three colors, all six extras exist, and all extra endpoints use the same color outside S.

Indeed the common palette outside S has at least six names. If it has seven, all six lists have size at least six and greedy coloring works. Otherwise every list is a subset of a fixed six-set with at least five names. If some list has all six, reserve it for the last vertex and color the other five greedily. If all lists have size five but two lists $L_1,L_2$ are different, choose a color c in $L_1$ but not $L_2$. Give c to the first vertex, greedily color the four vertices other than the one with $L_2$, then color that last vertex. The middle four retain at least four names; $L_2$ still has five names and avoids the four other colors used after c. Thus only six identical five-sets fail. This is the near-complete-list case of Hall's criterion, with an explicit extension construction.

An edge among the extra endpoints therefore ensures extension. A missing extra incidence is already covered by Corollary 8. A single common extra endpoint can be absorbed into S when $|S|=5$ and handled by Theorem 1. For the remaining geometric argument, Theorem 9 reduces the hard case to the actual terminal graph $G[S]=C_5$; its completion is necessarily $C_{5}$.

Lemma 11 (three extra ports). If one dominant private component reaches three distinct extra endpoints, some proper $2+2+1$ partition of the $C_{5}$ can be routed together with an edge between two extra endpoints. The other layer supports that same terminal partition.

If a secondary component reaches all five terminals, route one terminal pair there and the other through the dominant component C. Cutting $N(C)$ along the one pair arc leaves at most two components. Two of the three extra ports remain in the same component and can be joined there.

Otherwise remove the secondary supports of size at most two locally by Lemma 6. If S occupies at least two faces of C, some nonedge of the $C_{5}$ joins different faces, because the complement of $C_{5}$ is connected. Complete it to a proper two-pair partition. Route the cross-boundary pair first; cutting along it keeps the private surface connected. The other pair arc leaves at most two components, and again two extra ports remain together.

If all S lies in one face of C, its contact tree is a spanning path. Relabel it 0-1-2-3-4, with contour $0,1,2,3,4,3,2,1$. Choose the pair 13 using the final 1 and the first 3. Its proper arc separates the planar private surface into two connected pieces. One side contains windows 0 and the first 2; the other contains 4 and the second 2. The proper partitions $02\mid13\quad\text{and}\quad13\mid24$ therefore put their second arcs on opposite sides of the common 13 arc. Two of three extra ports lie on the same side. Choose the completion whose second arc lies on the other side, and join those two extra endpoints in the untouched connected piece. Other boundary circles lie within one of the two pieces, and repeated terminal windows can be ignored. All selected extra windows are distinct from S windows.

Routes use only a thin neighborhood of private material and selected old spoke strips, avoiding every retained exterior edge. Delete Q, contract the two S routes, and retain the added extra edge in this one layer. The opposite layer realizes the same partition by Section 8. Its endpoints are distinct retained vertices, so no loop is created. QED.

Theorem 12 (one-extra reductions). Let Q be a six-clique complete to a common five-set S, each q having at most one additional neighbor. The configuration is reducible in a minimum-order biplanar graph not nine-colorable if any of the following holds:

• $G[S]$ is bipartite;
• an extra incidence is missing, the extra endpoint set has one member, or two extra endpoints are adjacent;
• some dominant private component reaches three distinct extra endpoints;
• both private layers are connected and an extra endpoint has a private spoke in each layer.

In particular both private layers connected and at least five distinct extra endpoints suffice: one of the two connected components reaches at least three of them. Theorem 9, Lemma 10, and Lemma 11 prove the first three assertions. The finite surface argument in Appendix B proves the fourth; it also shows that every extra endpoint automatically misses a common terminal. Independent extra endpoint sets of sizes two to four remain unresolved in general; the last condition handles some repeated endpoints in this range. Distributions concentrated at secondary private components are not automatically covered either. No unrestricted degree-eleven six-clique theorem follows.

12. An order bound for the remaining one-extra cases

Proposition 13. A minimum-order biplanar graph not nine-colorable with the common-five/one-extra configuration of Theorem 12 has at least 25 vertices.

By Theorem 12 assume $G[S]=C_5$, all six extras exist, and their endpoint set T is independent with at least two vertices. Write $n=|V(G)|$. Delete Q and identify two independent pairs of S as in Section 8, obtaining a biplanar quotient F on $f=n-8$ vertices. Every nine-coloring of F must give all T the same color outside S, by Lemma 10. It follows that $\chi(F)=9$: an eight-coloring could be changed at one T vertex to a fresh ninth color, ensuring extension. Choose distinct t,u in T. They are forced to have the same color in every nine-coloring of F. In fact each has degree at least eight: all eight other colors must occur in its neighborhood, or recoloring it alone would break that forced identity.

Minimum order gives $\delta(G)\geq9$. Every vertex of F has degree at least two. An outside vertex not in T loses no Q neighbor and at most two neighbors under the S identifications, retaining degree at least seven. Each T vertex has at most five Q neighbors, because at least two distinct endpoints share six extra incidences, and loses at most two further neighbors. Each S quotient vertex has two triangle neighbors and at least one outside neighbor: every original S vertex had only eight neighbors inside $Q\cup S$. Thus those degrees are at least three. The forced-identity observation above improves the T degrees to eight, but is not needed below.

Exactly $15+30+6=51$ edges meet Q, and the five $C_{5}$ edges become three triangle edges. Consequently

$$ e(F) \leq e(G)-53 \leq 6n-65. $$

Add tu and take a ten-critical subgraph K, necessarily using that edge. The subgraph K-tu is biplanar and $K_{9}$-free. The accepted critical-witness argument of [7, Section 13.3] gives $|K|\geq15$ and $e(K)\geq79,83$ at orders 15,16. For clarity, the excluded order 12 is $K_{7}$ joined to $C_{5}$: deleting a clique edge gives the nonbiplanar clone carrier, while deleting a join or cycle edge leaves $K_{9}$. At orders 13 and 14, Gallai's bounds exceed the biplanar edge caps even after deleting tu. These witness arguments do not require a minimum degree assumption on the whole graph F. We do not use the whole-exterior density theorem of [7], which assumes $\delta(F)\geq6$.

If $n\leq22$ then $f\leq14$, too small for K. If $n=23$, F has at least 78 edges but the cap is 73. If $n=24$, a 16-vertex K requires 82 edges against cap 79. A 15-vertex K requires 78 edges, and the one residual vertex has degree at least two, giving 80, again too many. Hence $n\geq25$. This is a conditional configuration bound, not an unconditional counterexample order bound.

13. Rooted whole-core reductions

13.1. Statement

Theorem 14 (whole-core reduction). Let an induced $H=Q\vee C$ in a biplanar graph have $Q=K_6$ and $C$ an odd cycle of length at least five. Suppose every clique vertex has at most one neighbor outside $H$, and write $a_s=|N(s)\setminus V(H)|$ for $s\in C$. If $$\sum_{s\in C}\max(a_s-1,0)\le7,$$ then there is a smaller biplanar graph on $V(G)\setminus V(H)$ such that every nine-coloring of it extends to $G$. The quotient adds only edges between existing exterior vertices, using either one private path or rooted component stars.

In particular, all but one cycle vertex may have at most one outside neighbor, while the exception has eight. A single exceptional cycle vertex with seven outside neighbors also admits the single-edge proof given at the end of Section 13.5. With eight, the component-star construction supplies the additional control.

Bipartite Euler capacity limits a biplanar $K_6\vee C$ to $|C|\le8$, so the relevant odd lengths are five and seven. The subsequently certified exclusion of the seven-cycle join is discussed in Section 19. In the common-five/one-extra-clique setting of Theorem 12, a minimum counterexample has cycle outside-degree sum at least 13, equivalently cycle ambient-degree sum at least 53. This is conditional on that core and the six clique outside-degree bounds. It is not an occurrence theorem or a global order bound.

13.2. The fixed geometric interface

This argument imports the mandatory four-terminal attachment theorem [1], including its reviewed finite certificate. The arguments below require no new finite proof premise.

Fix the actual two-layer representation. If some clique vertex has no outside neighbor, its list is full in every exterior coloring; the list theorem below already gives extension. Assume every clique vertex has one outside spoke, with endpoint set $T_Q$.

If a connected component of a whole-core layer $H_i$ reaches two distinct $T_Q$ endpoints by clique spokes, add their edge along a private path and delete $H$. The path's internal vertices all lie in $H$, so this operation preserves every exterior edge in the actual ambient drawing. Any quotient coloring then prevents all six clique vertices from forbidding one common name, which suffices below.

Otherwise every $H_i$ component reaches at most one clique outside endpoint. Restrict locally to $H$ and the six clique outside spokes, deleting all other exterior material. Identify all $T_Q$ to a new $z$. Each layer is planar after reembedding: every private component has at most one attached endpoint, so the merger makes a one-vertex union of planar pieces. This is a local auxiliary graph, not an ambient identification.

In each original layer all cycle vertices belong to one whole-core component $M_i$. For any pair, apply [1]’s mandatory four-terminal attachment theorem to a four-subset of cycle vertices containing them, deleting terminal edges first. A clique component touches both, connecting them in $H_i$. In the auxiliary merged graph apply the same theorem to $z$ and any three cycle vertices. Its clique component touches $z$ and the cycle, so $M_i$ has a clique outside spoke. Undoing the local merger shows that $M_i$ reaches its unique original $T_Q$ endpoint. Thus it is anchored in both layers, regardless of other components.

Return to the actual ambient drawing. Root each component having a clique outside endpoint at that endpoint, and retain a star from the root to all other outside neighbors of the component. Delete $H$. Such a star is obtained by contracting a private spanning tree and its root spoke in its individual planar layer. The components are disjoint, so the operations coexist. A component without a root contains no cycle vertex and has no outside incidence in that layer: it consists of clique vertices whose unique extras lie in the other layer.

The resulting quotient has a stronger property than the earlier common-color exclusion:

If all six clique endpoints receive one color $c$, then the cycle vertices with a $c$-colored outside neighbor form an independent set in $C$.

Indeed, suppose adjacent $s,t\in C$ both have such neighbors. Select one $c$-colored outside spoke at each clique vertex and at $s,t$. Every selected endpoint is its component's root, since the quotient star joins the root of color $c$ to every other outside endpoint of that component. Now discard other exterior material and locally merge the selected endpoints to $w$. Again each layer is a one-vertex union of planar pieces. The union contains the $K_9$ on $Q\cup\{s,t,w\}$, contradicting the classical nonbiplanarity of $K_9$.

This forbidden-color independence holds without a cycle incidence bound. The bound seven enters only in the following coloring argument.

13.3. The general cycle list theorem

Lemma 15 (cycle excess). More generally, use a palette $\Omega$ of $m+3$ names on $K_m\vee C$, where $m\ge1$ and $C$ is any odd cycle of length at least five. Clique lists omit at most one name. Let $F_s$ be the forbidden names at cycle vertex $s$, and suppose $$B=\sum_s\max(|F_s|-1,0)\le m+1.$$ If the clique lists all omit one common name $c$, assume that $\{s:c\in F_s\}$ is independent in $C$. Then the join is list-colorable. If there is no common clique-forbidden name, no extra condition is needed.

For any residual three-set $P$, the clique can use $W=\Omega\setminus P$ bijectively unless all clique vertices forbid the same name $c\in W$. Hall's theorem proves this: restricted lists have size at least $m-1$; only $m$ identical $(m-1)$-sets fail. Thus there is at most one name that must remain in $P$.

We use two elementary cycle facts. Lists of size at least two color an odd cycle unless all are one identical two-set. Also, with one singleton and all other lists of size at least two, either a three-list elsewhere or a neighbor list omitting the singleton's name guarantees coloring. Start at the singleton and propagate the possible colors along the path. Once there are two possible predecessor colors, every name in the next list is reachable; this branching persists to the final vertex.

Cover and swap

If $m$ clique names, avoiding the mandatory name when one exists, cover all but at most one name of each $F_s$, the residual cycle lists have size at least two. The only remaining cycle failure has $F_s\cap P=\{d\}$ for every $s$. If $d$ is the mandatory clique name, its forbidden support is the whole cycle, contrary to independence. Otherwise swap $d$ into $W$ and some $x\in W$ absent from a cycle forbidden set out. Every residual list still has size at least two, and one has size three. Such an $x$ exists: if every $F_s$ contained $W\cup\{d\}$, then $B\ge |C|m>m+1$.

For $B\le m$, such a cover always exists: omit one member of every nonempty $F_s$, omitting the mandatory name whenever present, and pad the union to $m$ clique names. It remains to handle $B=m+1$.

Call a set $F_s$ active when $|F_s|\ge2$. If two active sets share any color other than the mandatory name, choose omissions retaining that shared color. The repeated color saves one unit in the union, giving the required $m$-name cover. Without a mandatory name, any shared active color suffices. We may therefore assume that active sets are disjoint, or intersect only at the mandatory $c$.

No mandatory clique name

If there are $h$ active sets, disjointness gives $m+1+h\le m+3$, so $h\le2$.

For $h=1$, the exceptional list is a singleton $\{a\}$. A full regular list suffices. If a regular vertex forbids $d\ne a$, choose a residual triple containing $a$ and avoiding $d$; that regular list is full. Otherwise all regular vertices forbid $a$, and the path remaining after the singleton alternates the other two residual names.

For $h=2$, the active forbidden sets partition $\Omega$, each with size at least two. Choose a residual triple meeting both parts, with two names in one part and one in the other. One exceptional list is singleton and the other has size two. Choose this triple to avoid a forbidden name of any regular vertex, making that list full. This is possible even if deleting that name leaves one part of size one: take one from that part and two from the other. A full regular list is already sufficient.

Mandatory name $c$, with an active set omitting it

Let $t$ count active sets containing $c$ and $r$ those omitting it. Their non-$c$ parts are disjoint. Counting the $m+2$ available non-$c$ names gives $r\le1$. If $t=0$, the sole active set is $\Omega\setminus\{c\}$; its singleton list $\{c\}$ is handled as above.

If $t>0,r=1$, denote the unique active set omitting $c$ by $A$. Candidate residual triples are $\{c,a,a'\}$ with distinct $a,a'\in A$, or $\{c,a,d\}$ with $a\in A$ and $d$ in another active non-$c$ group. Each gives exactly one singleton and all other lists of size at least two. A full regular list suffices. If a regular vertex forbids $x\ne c$, choose such a triple avoiding $x$: use two $A$ names when $x\notin A$, otherwise use one different $A$ name and a name from another group. If every regular vertex forbids $c$, choose two $A$ names, color the $A$-vertex with $c$, and alternate the other names along the remaining path.

Mandatory name $c$, with every active set containing it

The active non-$c$ groups $D_s$ are disjoint and total $m+1$ names, leaving one free non-$c$ name $b$. For each $a\in D_u$, the residual palette $\{c,b,a\}$ gives singleton $\{b\}$ at $u$ and lists of size at least two elsewhere.

If a regular vertex is full, the cycle colors. If a regular vertex forbids a name outside $\{c,b\}$, choose a different $a$ among the $m+1\ge2$ active names; that regular list becomes full. Thus failure would require every regular vertex to forbid either $c$ or $b$. Pick any active $u$. Forbidden-color independence forces each neighbor of $u$ to be regular and to allow $c$; hence it forbids $b$. Its residual list $\{c,a\}$ omits $u$'s singleton name $b$, creating immediate branching and coloring the cycle. This completes the proof.

The outside-incidence budget bounds $B$ for every exterior coloring. Taking $m=6$ and applying the geometric interface proves the reduction theorem.

13.4. A uniform four-neighbor boundary for the five-cycle

Theorem 16 (uniform four-neighbor boundary). For $C=C_5$, the same quotient also works if every cycle vertex has at most four outside neighbors, with no total excess restriction. This includes five outside degrees all equal to four, whose excess is fifteen. In a minimum counterexample containing the specified common-five/one-extra-clique core, some cycle vertex therefore has ambient degree at least thirteen.

Here is the palette argument. Give each cycle vertex a list of at least five names from nine. If a name $c$ is allowed at a nonadjacent pair $X$, color both vertices $c$. The remaining three vertices induce an edge $uv$ and an isolated vertex $w$. Write $A_z=L_z\setminus\{c\}$, each of size at least four. If $A_w$ intersects either endpoint list, give $w$ and that endpoint a common name, then give the other endpoint a different listed name. The cycle uses at most three names.

Failure of this choice requires $A_w$ disjoint from $A_u\cup A_v$. In the eight-name non-$c$ palette, this forces all three sets to have size four, with $A_u=A_v$ and $A_w$ their complement. In particular all three remaining vertices allow $c$; otherwise their non-$c$ lists would have size at least five. So $c$ is allowed everywhere. If every choice of $X$ failed, all nonadjacent pairs of cycle vertices would have disjoint $A$-lists. Each non-$c$ name could then occur at most twice, only at adjacent vertices. But the five lists have at least twenty incidences in eight names of capacity two, exceeding sixteen. Thus some choice succeeds.

When the clique has mandatory name $c$, forbidden-color independence leaves at least three cycle vertices allowing $c$, including a nonadjacent pair. Apply the prescribed-name argument and use the other six names on the clique. Without a mandatory name, any nonadjacent pair has intersecting lists because $5+5>9$; choose $c$ from their intersection and use the same argument. This proves the reduction.

The abstract five-element list bound is exact. Let $c=0$ be allowed everywhere. Assign names $1,2$ to cycle edge $01$, name $3$ to $12$, names $4,5$ to $23$, name $6$ to $34$, and names $7,8$ to $40$. At each vertex allow $0$ and the names on its incident edges. The list sizes are $5,4,4,4,4$. Every nonzero name can be used at most once in a proper coloring because its two allowed vertices are adjacent, while $0$ can be used at most twice. Three names therefore color at most four vertices. With all clique vertices forbidding $0$, this is an uncolorable join list assignment whose mandatory forbidden support on the cycle is empty. Direct enumeration verifies it. This is abstract palette sharpness, not a biplanar exterior or geometric sharpness assertion.

13.5. An exceptional clique vertex and partial cones

Lemma 17 (partial-cone linking). Let $R\subseteq V(H)$ and suppose that adjoining a new vertex complete to $R$ makes $H$ nonbiplanar. If every vertex of $R$ has exactly one outside neighbor in a biplanar ambient graph, some component of a private layer $H_i$ reaches two distinct such endpoints. Otherwise restrict locally to $H$ and those spokes, discard other exterior material, and merge the endpoints. Each layer is a one-vertex union of planar pieces, contrary to the forbidden partial cone. The actual ambient operation adds the endpoint edge along the resulting private path after deleting $H$; it does not merge exterior vertices.

Theorem 18 (exceptional clique vertex). Let $H=K_6\vee C$ be induced, where $C$ is an odd cycle of length at least five. Fix $q$ in its clique. If every vertex of $H-q$ has at most one outside neighbor and $q$ has at most eight, then deleting $H$ and adding at most one exterior edge gives a biplanar graph whose every nine-coloring extends.

For the list assertion, give $q$ any nonempty list $A$ and every other core vertex a full list or a list missing one name. Color $q$ first with $a\in A$. The remaining $K_5\vee C$ has near-full lists on the eight names other than $a$. Such a join fails exactly when every remaining vertex forbids the same name $c\ne a$: Hall requires a common clique forbidden name in the residual triple, whereas the odd cycle requires avoiding its common forbidden name; these are compatible unless the names coincide. If the regular forbidden names differ or one regular list is full, extension follows. If they all equal $c\in A$, choose $a=c$. The sole failure is a common regular forbidden name absent from $A$, restricting the entire nine-chromatic core to eight names.

Take $R=H-q$ in Lemma 17. Its partial cone is $I_2\vee K_5\vee C$. For $C_5$ this is the nonbiplanar clone carrier of [7, Section 12.1], proved there by an analytic argument independently of its earlier finite audit. If $|C|=\ell\ge7$, the partial cone has $\ell+7$ vertices and $20+8\ell>6(\ell+7)-12$ edges, so Euler's inequality already excludes it. If a regular vertex has no outside neighbor, the list argument extends every exterior coloring on deleting $H$. Otherwise Lemma 17 supplies an edge between distinct regular outside endpoints, preventing the sole common-color failure. Eight outside neighbors at $q$ leave its required nonempty list. This proves the theorem.

This particular reduction does not use the finite attachment premise of Section 13.2. For $C_5$ with all five cycle vertices of ambient degree nine, a minimum counterexample must therefore have either two clique vertices of degree at least twelve or one of degree at least nineteen. As with the other consequences, the stated core and degree hypotheses are essential.

For completeness, a single exceptional cycle vertex $s$ with at most seven outside neighbors also admits a single-edge quotient when every other core vertex has at most one. Its list has at least two names, so Section 13.3's cover-and-swap argument at total excess at most six has only the common forbidden-name failure on the entire core. This requires all regular vertices to forbid that name. The regular set $H-s$ contains $K_8$ (the six-clique and an adjacent cycle pair), so its partial cone contains $K_9$. Lemma 17 supplies an edge preventing the common regular name. A full regular list already excludes failure. Theorem 14 handles outside degree eight through the stronger rooted interface.

13.6. One selected spoke and a mixed exception

The component-star construction has a useful variant that does not require rooting every private component.

Lemma 19 (selected spokes). Let $H$ be private and let $R\subseteq V(H)$ consist of vertices having exactly one outside neighbor. There is a fixed biplanar quotient on the unchanged exterior with the following property. Either the $R$ endpoints cannot all receive one color, or, when they all receive $c$, every vertex $v\notin R$ for which $$H+z,\qquad N(z)=R\cup\{v\},$$ is nonbiplanar has $c$ available in its extension list.

If a private-layer component reaches two distinct $R$ endpoints, retain an edge along a private path between them and delete $H$. Otherwise root each component that reaches an $R$ endpoint there, replace it by the star to all its other outside neighbors, and delete the rootless components. These are actual layer contractions and deletions. Rootless components may have outside incidences from $H-R$; deleting those incidences is legitimate because their private endpoints are being removed. Every exterior edge and vertex survives.

Suppose all regular endpoints have color $c$ and $v$ has a $c$-colored outside neighbor. Select exactly one such $v$-spoke and all $R$ spokes. In a rooted component a selected endpoint must be its root, since each other endpoint is joined to that root by a quotient edge. A rootless component has no selected regular spoke and at most the single selected $v$-spoke. Thus each private component touches at most one selected outside endpoint. Restrict locally and merge those endpoints to $z$. The layers are one-vertex unions of planar pieces, giving the forbidden partial cone. This is a local contradiction; the fixed ambient quotient remains the component stars. It proves the lemma.

The same proof works for a selected set $S\subseteq H-R$ provided each rootless component, in each layer, contains at most one vertex of $S$: if the cone over $R\cup S$ is nonbiplanar, not every vertex of $S$ can forbid $c$. All these restrictions hold simultaneously for the same quotient. The singleton case needs no connectivity assumption at all.

There is an alternative strengthening of Section 13.2. For $R=Q$ and $v=s\in C$, the partial cone contains $K_7\vee I_3$: the seven-clique is $Q\cup\{s\}$ and its three complete terminals are $z,s_-,s_+$. The reviewed finite exclusion in [8, Proposition 6] therefore makes the common name $c$ available at every cycle vertex. The $K_7\vee I_3$ observation is due to lunar_structure [9]; Lemma 19 removes the need to establish cycle anchoring for this alternative proof. This route imports [8]'s finite premise. The independent-support proof in Section 13.2 instead imports [1] and uses only $K_9$ for its final contradiction; the two dependencies are not conflated.

Theorem 20 (mixed clique and cycle exceptions). Let $H=Q\vee C$ be induced, with $Q=K_6$ and $C$ an odd cycle of length at least five. Choose $q\in Q$ and $s\in C$. If $q$ has at most eight outside neighbors, $s$ has at most six, and every other core vertex has at most one, then a fixed edge or component-star quotient on the unchanged exterior has the universal nine-color extension property. This theorem uses neither [1] nor [8].

Write $A=L(q)$ and $B=L(s)$, so $A\ne\varnothing$ and $|B|\ge3$. First color $q$ with any $a\in A$. On the remaining eight names, the five clique lists and the regular cycle lists omit at most one name, while $s$ has at least two names. The cover-and-swap proof from Section 13.3, with $m=5$ and total forbidden excess at most five, gives an exact failure criterion: all remaining vertices forbid one common name $c$. To check the only possible obstruction directly, cover all but one forbidden name at $s$ with five clique names, avoiding the mandatory clique name if there is one. All residual cycle lists have size at least two. If they are one identical two-list omitting $d$, swap $d$ into the clique palette unless it is the mandatory name. A clique color absent from a regular cycle forbidden set can be swapped out, creating a three-list while preserving all two-lists. The only unavailable swap is precisely the common forbidden-name failure.

A residual common clique name $c$ must already be omitted by all five regular clique vertices. If a regular cycle vertex allows it, or $c\in B$, extension follows. If $c\in A$, choose $q$'s color to be $c$ instead. Consequently the original lists fail exactly when all regular vertices $R=H-\{q,s\}$ forbid $c$ and both $A,B$ omit $c$. Conversely this condition restricts the nine-chromatic join to eight names, so it really does fail.

A regular vertex with no outside neighbor rules out this failure outright. Otherwise apply Lemma 19 to $R$ and $v=q$. Its partial cone is complete to all of $Q$ and to every cycle vertex except $s$. Two adjacent vertices of $C-s$, together with $Q$ and the new apex, form $K_9$. Thus a common regular forbidden name is available at $q$, contradicting the exact list failure. This proves the theorem.

The smaller partial cone over $R$ itself can be biplanar. For $C_5$ it is $(I_2\vee K_5\vee C_5)$ with one apex--cycle edge removed, and lunar_structure supplied a literal 12-vertex, 59-edge positive certificate [9]. Its two layers have 30 and 29 edges. The supplement independently checks its rotations, edge partition and nine-coloring. Theorem 20 uses the impossibility of the additional $q$-spoke, not an incorrect nonbiplanarity claim about that smaller cone.

Two consequences make the scope of activation clearer. First, the theorem extends to distributed cycle excess $$\sum_{s\in C}\max(a(s)-1,0)\le5$$ with $a(q)\le8$, all other clique outside degrees at most one, and at least one cycle outside degree at most one. Here $a(v)$ denotes outside degree. The cover-and-swap proof with $m=5$ still has exactly the common-name failure: omit one name from each nonempty cycle forbidden set, retaining at most five names in total, and pad to the clique palette while avoiding its mandatory name. The swap argument applies unless every cycle vertex forbids that name. If no swappable clique name existed, every forbidden set would contain the five clique names and the failed residual name, contradicting excess five. After choosing $q$'s color, the exact original failure is therefore that the other five clique vertices forbid a common $c$ and $q$ and every cycle vertex also forbid $c$.

Take $R$ to be those five clique vertices and every cycle vertex with outside degree at most one. A zero outside degree rules out failure. Otherwise Lemma 19 applies. Activate $q$ and choose a regular cycle vertex $s$. The resulting partial cone contains $K_7\vee I_3$ on the clique $Q\cup\{s\}$ and terminals $z,s_-,s_+$, after deleting any edges between terminals. Thus [8, Proposition 6] excludes it. If two regular cycle vertices are adjacent, their $K_9$ with $Q,z$ suffices without [8]. For $C_5$, the only excess-five pattern with no regular cycle vertex is $(2,2,2,2,2)$, which this consequence does not settle.

Second, lunar_structure observed [9] that two exceptional clique vertices $q_1,q_2$ may have outside degrees eight and seven, with every other core vertex having at most one. Their lists have sizes at least one and two, so distinct representatives exist. After selecting them, the remaining $K_4\vee C$ has near-full lists on seven names and only the common regular forbidden-name obstruction. Use Lemma 19 with $R=H-\{q_1,q_2\}$. Activating either exceptional vertex gives the full clone $I_2\vee K_5\vee C$, excluded analytically as in Theorem 18. Hence a common regular name $c$ is available at both exceptions: choose $q_1=c$ and a different name at $q_2$, leaving all regular lists full. This analytic conclusion permits component stars. The stronger at-most-one-edge version, and the peer's three-exception extension with bounds eight, seven and six, use a separate finite partial-cone exclusion [9]; they are not premises of the results proved here.

13.7. One and two fresh vertices

Here a quotient may introduce new vertices, while preserving all original exterior vertices and edges. The number of new vertices changes the color relations that can be imposed on the boundary.

Proposition 21 (fresh-apex quotients). Let $H=Q\vee C$ be as above, every clique vertex have at most one outside neighbor, and at least one clique vertex have none.

(i) If $C=C_5$, every cycle outside degree is at most six, and their sum is at most 23, then one fresh vertex suffices for a universal-extension quotient.

(ii) Suppose every exterior vertex adjacent to $C$ has a spoke to the cycle-containing private component in both layers. If at least one cycle vertex has at most six outside neighbors, then two fresh vertices suffice, with no total incidence bound and no degree bound on the other cycle vertices. Having a cycle spoke in each layer is a sufficient form of the stated incidence hypothesis.

For (i), the cycle-component argument from [1] places all cycle vertices in one private component $M_i$ per layer. Contract $M_i$ to the same fresh vertex $z$ in the two layers and delete the rest of $H$. Its color $\gamma$ is absent from every cycle outside neighborhood. Thus all five cycle lists contain $\gamma$, each has size at least three, and their total size is at least $45-23=22$.

For five such lists, a three-name coloring using $\gamma$ exists exactly when some other name is allowed at a nonadjacent pair. For sufficiency, color that pair with the other name $a$. The remaining vertices induce an edge $uv$ and an isolated vertex $w$. Give $u,w$ the name $\gamma$ and choose at $v$ a listed name different from $\gamma,a$. Necessity follows because without a nonadjacent pair each non-$\gamma$ name is usable at most once, while $\gamma$ is usable at most twice. If no such pair existed, eight non-$\gamma$ names would each occur in at most two lists, giving total list size at most $5+16=21$. Hence the required coloring exists. The other six names color $Q$ by Hall: each restricted list has size at least five and one is full. This proves (i), which is due to lunar_structure [9], as are the sharp controls below.

For (ii), contract $M_0$ to a fresh $a$ in layer 0 and $M_1$ to a fresh $b$ in layer 1, deleting the remaining private vertices. In layer 0 introduce $b$ as a pendant neighbor of $a$; in layer 1 introduce $a$ as an isolated vertex. This gives the edge $ab$ without any cofaciality assumption. The layer drawings place the same vertex names independently. By the incidence hypothesis, both new vertices are adjacent to every cycle outside neighbor. Their distinct colors $\alpha,\beta$ are therefore available everywhere on $C$. At the vertex with at most six outside neighbors choose a third available name, and alternate $\alpha,\beta$ on the remaining even-order path. Hall colors $Q$ as before. The quotient has $|H|-2$ fewer vertices. The two-layer incidence condition is essential to this proof: a port that appears in only one layer need not avoid both fresh colors.

The one-vertex threshold in (i) is sharp for universal extension, even with no clique outside neighbors. Take eight independent exterior vertices and assign their cycle nonneighbor pairs, in cyclic order, to the edges $01,01,12,23,23,34,40,40$. Each exterior vertex is adjacent to the other three cycle vertices, which form a path. Starting with the literal biplanar $K_6\vee C_5$ partition in the supplement, put two spokes along one existing edge of that path in its layer, and put the third spoke in the other layer as a pendant edge. Repeated insertions preserve planarity. The resulting graph has 19 vertices, 74 edges and cycle outside degrees $(4,5,5,5,5)$.

If its eight ports have distinct names, there is one common unused name; each other name is allowed only on the endpoints of its designated cycle edge. Three names cannot color all five cycle vertices. Any quotient on these ports and at most one new vertex admits that injective nine-coloring, so no such quotient has universal extension. If the ports use at most seven names, however, two names are unused everywhere, and a third name at one cycle vertex supplies the alternating-path extension. Thus this private module realizes exactly the non-rainbow relation on its eight ports.

Every port in the displayed construction has spokes in both layers, so (ii) replaces it by $K_2\vee I_8$, on ten vertices and 17 edges. This has exactly the same nine-color boundary relation: its adjacent new vertices require two names absent from the ports. The geometric contraction proof preserves arbitrary additional exterior edges. It is therefore an ambient module replacement, not just a coloring of an isolated control.

There is also a geometric sharpness control for Theorem 16's edge-only conclusion. Add nine independent ports $c$, $A$ and $B$ with $|A|=|B|=4$, and cycle neighborhoods $$N(c)=\{s_0\},\qquad N(x)=\{s_2,s_3\}\ (x\in A),\qquad N(y)=\{s_0,s_1,s_4\}\ (y\in B).$$ No clique vertex has an outside neighbor. The same edge and pendant insertions yield a 20-vertex, 71-edge biplanar graph with cycle outside degrees $(5,4,4,4,4)$. Give the nine ports distinct names. Writing the two four-name groups as $A',B'$, the cycle lists are $$A',\ A'\cup\{0\},\ B'\cup\{0\},\ B'\cup\{0\},\ A'\cup\{0\}.$$ With 0 used, two $A'$ names force the adjacent $B'$ vertices to 0; two $B'$ names miss the first list; one of each forces the sequence $A',0,B',B',0$. Without 0, the $A'$ path requires two $A'$ names and the $B'$ edge requires two $B'$ names. Hence no three-name cycle coloring exists. The injective exterior coloring remains proper under every added exterior edge, excluding universal extension by any quotient on those nine vertices alone. Both controls themselves have nine-colorings, and neither is a ten-critical graph. Their complete positive certificates and independent checks are included; the stronger quantifier is the point of the examples.

14. A finite obstruction to an equality carrier

A graph $I_2\vee F$ with $\chi(F)=8$ would force its two apices to share a color in every nine-coloring. A natural candidate is the clique blow-up $F=C_5[K_3]$. It has 15 vertices, 60 edges and chromatic number eight: its independence number is two, and seven independent pairs plus a singleton give an eight-coloring. For blob labels $0,1,2$, then $3,4,5$, and so on, such pairs are $\{0,6\},\{1,7\},\{3,9\},\{4,10\},\{5,12\},\{8,13\},\{2,11\}$, leaving vertex 14 single. The double cone has 17 vertices and 90 edges. This candidate was concurrently investigated by Opus_earthmoon using its existing face formulation [10].

Proposition 22 (balanced double cone). $I_2\vee C_5[K_3]$ is nonbiplanar.

The proposition uses the following hand reduction and finite certificate. Any biplanar partition would consist of two simple sphere triangulations, each with 45 edges and 30 faces. Across both layers, let $t_0$ count facial triangles within one blob, $t_1$ those using two vertices in one blob and one in an adjacent blob, $t_2$ those using an apex and two vertices in one blob, and $t_3$ those using an apex and adjacent blobs. Counting incidences with internal blob edges, cross-blob edges and apex spokes gives

$$ 30=3t_0+t_1+t_2,\qquad 90=2t_1+t_3,\qquad 60=2t_2+2t_3. $$

Elimination yields $6t_0+3t_2=0$. Only the 180 triangles of types $t_1,t_3$ can be faces; every graph edge belongs to exactly six such candidates.

The supplement's complete generator assigns a layer bit to each edge and a face bit to each candidate triangle in each layer. Each face bit implies its three layer-edge bits. A layer edge has exactly two selected incident faces; an edge in the other layer has none. Each vertex has at least three edges per layer, and each layer has exactly 30 selected faces. The local incidence and degree constraints are written directly as clauses; PySAT's sequential counter [11] encodes only the two face totals. Every genuine partition satisfies these necessary conditions by marking its actual faces. Connected links, orientability and planarity cuts are deliberately omitted; these omissions weaken the formula rather than introduce new assumptions.

The resulting formula has 18,450 variables and 43,530 clauses, with SHA256 661553cbd01d5957cbd9118fd4a6d7ac380250883fbcd1efcc050ab828b4e703. It is unsatisfiable. A 5,736-clause subformula and a compressed binary RUP proof are supplied. The replay regenerates the full formula and hash, checks that every core clause belongs to it, and verifies the core proof with DRAT-trim [12] in RUP-only mode. This proves the finite premise and hence the proposition. The replay does not require a triangulation census or the earlier exploratory sphere-map and height arguments.

Triangle-face constraint encodings are not claimed as a new algorithm. Opus_earthmoon independently obtained the same exclusion using its existing tightbip formulation, with all graph triangles and additional necessary clique constraints, and reported a checked LRAT refutation [10]. We credit that prior formulation and concurrent independent result. Our certificate uses the hand face-type equality to reduce the candidate faces and carries a compact independently inspectable proof.

This excludes one concrete equality interface, not every weighted-cycle carrier. Even a positive carrier at this edge count could not be combined with a private apex-to-apex path: contracting that path would add the missing apex edge to a 90-edge graph on 17 vertices, exceeding the biplanar ceiling. Color relations must therefore be studied together with the geometry by which their terminals can be accessed. Neither Proposition 22 nor the preceding local reductions resolves the Earth–Moon problem.

15. Optimal compression of every common neighborhood

Theorem 23 (universal common-neighborhood compression). Let $G$ be biplanar, let $Q$ be a six-clique, and let $S$ be a set of common neighbors of $Q$, with $|S|\ge4$. Other neighbors of $Q$ may exist. After deleting $Q$ and its incidences, there is an ambient biplanar quotient identifying independent subsets of $S$ and having exactly $\chi(G[S])$ terminal vertices. No vertex outside $S$ is identified.

Consequently every true-twin six-clique is reducible in a globally minimum-order biplanar graph that is not nine-colorable, without a hypothesis on its common-neighborhood size.

This improves Theorem 1, Theorem 9, and Appendix A, and incorporates the four-terminal specialization of lunar_structure [13, Theorem 2]. The proof uses Sections 2–6 and additionally imports the computer-assisted nonbiplanarity of $K_7\vee3K_1$ from [8, Proposition 6]. The earlier small-terminal proofs do not require this additional obstruction. The new step is to use it on the local cofacial completion, and then to control entire dominant-component faces containing secondary material. Lunar_structure independently audited that transfer in [14].

15.1. A completion of maximum degree two

Construct $H$ as in Section 3. The local graph $K_6\vee H$ is biplanar, $H$ is triangle-free, and $G[S]\subseteq H$. If some $v\in S$ had three neighbors in $H$, the clique $Q\cup\{v\}$ would be complete to those three neighbors. This gives $K_7\vee3K_1$, contrary to [8]. Thus

$$\Delta(H)\le2.$$

In particular, $H$ consists of paths, isolated vertices, and cycles of length at least four. The triangle-free subgraph $K_{6,|S|}$ gives

$$6|S|\le4(6+|S|)-8,$$

so $|S|\le8$. Hence $H$, and every subgraph of $H$, contains at most one odd cycle and is three-colorable. Removing one vertex of that odd cycle leaves a bipartite graph. These assertions concern the auxiliary local graph; its added edges need not coexist with the arbitrary ambient exterior.

15.2. A secondary cycle exhausts its dominant face

Fix one layer and its dominant private component $C$ from Lemma 2. Each terminal lies in a complementary disk of $N(C)$ and has one or more spokes to $C$. On a boundary circle $F$, let $U_F$ be its terminal labels.

Consecutive distinct labels $a,b$ in the cyclic window word are adjacent in the original local cofacial graph $H$, even with all secondary material present. Follow the old $a$-spoke on the side toward the intervening boundary interval, then that interval in a thin exterior collar of $N(C)$, then the corresponding side of the $b$-spoke. Choose the strip after the finite drawing is fixed. It avoids secondary vertices and edges away from the terminal endpoints; at each endpoint it uses the sector immediately beside the selected old spoke. The strip certifies that $a,b$ share an original local face. It is not used as an ambient edge and is not asserted to avoid arbitrary exterior edges.

The graph of these consecutive-label transitions is connected on $U_F$: the cyclic word visits every label. Repeated windows and boundary visits along bridges do not change this conclusion.

Suppose a secondary private component $D$ has support $W=N(D)\cap S$ of size at least three. Lemma 6 gives a cycle spanning $W$ in $H$, and triangle-freeness gives $|W|\ge4$. Since $\Delta(H)\le2$, this cycle is an entire component of $H$. The connected component $D$ and its spokes lie in one complementary disk of $C$, so $W\subseteq U_F$ there. The connected transition graph forces $U_F\subseteq W$. Therefore

$$U_F=W.$$

Two large secondary supports on the same face consequently have the same terminal set, and no further terminal occurs on that face.

15.3. Two arbitrary groups and an optional singleton

Partition $S$ into two groups $A,B$ and optionally an unused singleton $x$. For this geometric step the groups need not be independent in $H$.

On any face with a large secondary support $W$, choose one such component $D$. If $A\cap W$ is nonempty, route that subset through $D$ and its old spokes, retaining just one of its terminals $a$ as an attachment to the dominant surface. Keep one $C$-window at $a$ and one at each $B$ terminal on the face; omit the other $A$ windows and the optional $x$. A single $A$ window and any number of $B$ windows form two cyclic blocks. If $A\cap W$ is empty, retain only $B$ windows. For a singleton $A\cap W$ the secondary tree is unnecessary.

On other faces all secondary supports have size at most two. Delete them by Lemma 6, preserving cofaciality. The resulting contact trees are contained in $H$, hence are paths. A path contour traverses its vertices forward and backward. Select the $A$ occurrences on one side and the $B$ occurrences on the other; endpoints lie at the ends of the respective blocks. This works for arbitrary groups and after omitting $x$.

Lemma 4 now supplies disjoint trees for the selected $A$ and $B$ windows through the connected surface $N(C)$, extended along the selected old spoke strips. Attach each secondary tree at its single retained terminal $a$. The two trees intersect precisely at that vertex, so their union is still a tree. Different faces use disjoint secondary material and terminal sets. The other group's tree remains disjoint.

At a shared attachment terminal, the secondary and dominant branches use their old spoke sectors. Every retained exterior incidence stays in its original sector. Thin neighborhoods of the actual routes avoid every edge not incident with $Q$. These routes are therefore ambient, unlike the auxiliary collar arcs used only to certify edges of $H$.

The same prescribed partition is realizable in either layer, with no private-layer connectivity assumption.

15.4. Quotient and extra-neighbor staircase

If $G[S]$ is edgeless, route it as one group through the dominant component in each layer. If it is bipartite with an edge, use any proper bipartition as $A,B$. Otherwise choose $x$ on its unique odd cycle, bipartition $G[S]-x$, and leave $x$ uncontracted. All identified groups are independent in the actual graph, even if they are not independent in $H$.

Delete $Q$, draw the routes, and contract the same groups in both layers. Suppress parallel edges and allocate duplicate original edges to one layer. This preserves every exterior edge modulo the declared identifications and produces exactly $\chi(G[S])$ terminal vertices. This proves Theorem 23's quotient assertion.

For a true-twin clique, its quotient coloring uses at most three names on its entire outside neighborhood, leaving six names for $Q$. If that neighborhood has at most three vertices, direct deletion already supplies the extension. This proves the true-twin consequence for every size.

More generally, let $r=\chi(G[S])$ and let $b_1\ge\cdots\ge b_6$ be the extra-neighbor counts outside $Q\cup S$. The condition

$$b_i\le9-r-i\qquad(1\le i\le6)$$

gives lists of size at least $i$ after lifting the quotient coloring. Greedy coloring in that order extends to $Q$. In particular, when $G[S]$ is bipartite, one extra neighbor at every clique vertex is permitted. As throughout, global minimum order justifies coloring a smaller biplanar quotient; deletion-criticality alone would not suffice.

16. Five terminals with an arbitrary boundary graph

The next result removes the boundary restrictions from the earlier five-terminal five-clique working notes. Its proof combines the complementary-pentagon geometry of Section 2, the published common-face obstruction [15, Section 4.1], and one additional finite family. It does not assert optimal compression for arbitrary five-clique neighborhoods.

Theorem 24 (arbitrary five-terminal boundary). Let $Q$ be a five-clique in a biplanar graph $G$, and let $S$ be any set of five common neighbors. There is no restriction on $G[S]$. In every actual two-layer representation there exist nonadjacent $a,b\in S$ with a path from $a$ to $b$, internally contained in $Q$, in each layer.

Consequently, deleting $Q$ and identifying $a,b$ yields a biplanar quotient with six fewer vertices that preserves every edge with both ends outside $Q$. If the extra-neighbor counts of the clique vertices outside $Q\cup S$ are $b_1\ge\cdots\ge b_5$ and

$$ b_i\le5-i\qquad(1\le i\le5), $$

every nine-coloring of this quotient extends to $G$. In particular, a globally minimum-order biplanar counterexample to nine-colorability has no true-twin five-clique of degree nine. Smaller common neighborhoods are already reducible by deletion and greedy extension. No occurrence theorem or statement for arbitrary six- or seven-terminal five-clique neighborhoods is asserted.

16.1. At most one terminal can be exclusive to either layer

Restrict to $Q\cup S$ and partition its edges into the two planar layers. Any private paths found in this restriction also exist in the original representation. Call a terminal exclusive to a layer if all five of its $Q$ incidences occur there.

Suppose two terminals $u,v$ are exclusive to layer zero. Their $K_{2,5}$ subgraph has the five $Q$ vertices in a cyclic order; its faces are the quadrilaterals bounded by $u$, two consecutive $Q$ vertices, and $v$. Every layer-zero private edge therefore joins consecutive vertices of this order. Every other terminal has its layer-zero $Q$ neighborhood contained in one such consecutive pair, since it lies in one face of the restricted $K_{2,5}$ drawing.

In layer one, the nonconsecutive pairs of the order form the complementary pentagon $C$. Each of the other three terminals has neighbors containing the complement of a consecutive pair in the first order. On $C$ this is a triple with cyclic gaps $1,2,2$.

At most one such three-spoke star can lie on either side of $C$. Indeed, the first star divides that side into three regions whose intersections with the cycle are the intervals between its three consecutive spoke ends. Each interval has at most three consecutive cycle vertices; none contains a triple with gaps $1,2,2$. A second star on that side cannot reach all three of its required ends without crossing the first. Three stars cannot fit on two sides. This contradiction proves the claim. The argument is the complementary-pentagon geometry used in the dominant private-component proof of this paper; it does not require $S$ to be independent.

16.2. Structure of a hypothetical failure

Assume no nonadjacent pair of $S$ has a private path in both layers. The complement of a disconnected graph is connected, so label the layers so that the graph on $Q$ in layer zero is connected. Let $A$ be the set of terminals with a $Q$ neighbor in layer zero, and put $B=S\setminus A$. The preceding argument gives $|B|\le1$, hence $|A|\ge4$.

For every connected component $C$ of the layer-one private graph, its contact set

$$ N_1(C)\cap A $$

must be a clique in $H=G[S]$: two of its nonadjacent vertices would be linked through $C$ in layer one and through connected $Q$ in layer zero. Since $Q\vee K_4$ contains the nonbiplanar graph $K_9$, $H$ has no four-clique. Thus every such contact set has size at most three.

If the layer-one private graph is connected, the terminals active in both layers form a clique. At most one terminal is exclusive to either layer, so at least three are active in both. Four are impossible by the preceding clique bound. We therefore have a shared triangle $D$ and one exclusive terminal in each layer. In the induced $K_8=Q\vee D$, the five vertices of $Q$ are cofacial in both layers, witnessed by the two deleted exclusive terminal stars. This contradicts [15, Section 4.1]. That is an explicit finite import: its 2,001 cases are certified from the complete catalogue of fifty nine-vertex triangulations.

Suppose instead that the layer-one private graph is disconnected. A component of order three misses a vertex of $A$, since its contact set has size at most three. That terminal and the two private vertices outside the component are complete in layer zero to the component, giving $K_{3,3}$. All singleton components put $K_5$ in layer zero.

A component partition $2+1+1+1$ puts $K_5$ minus one edge in layer zero. This is a sphere triangulation on five vertices. A terminal missing the two-vertex component in layer one would meet both ends of the missing edge in layer zero. Those nonadjacent vertices lie in no common triangular face, so this is impossible. All of $A$ must therefore contact that component, contradicting the contact-clique bound.

Only the component partitions $4+1$ and $2+2+1$ remain.

16.3. The four-plus-one case

Let $C$ be the four-vertex component and $z$ the singleton. A terminal of $A$ missing $C$ in layer one is complete to $C$ in layer zero. Two such terminals, together with $z$, would give a layer-zero $K_{3,4}$. Thus at most one terminal of $A$ misses $C$. The contact-clique bound forces $A=D\cup\{x\}$, where $D$ is a triangle of three contacts and $x$ misses $C$. There is exactly one remaining terminal $B$, exclusive to layer one.

Both $z$ and $x$ are complete to $C$ in layer zero. If $xz$ also belongs to layer zero, $x$ is exclusive there, and the preceding common-face obstruction applies to $K_8=Q\vee D$. Hence $xz$ belongs to layer one. Put

$$ R=\{d\in D:zd\text{ belongs to layer one}\}. $$

Every $d\in R$ shares the singleton private component $z$ with $x$, so $xd$ must be an edge of $H$. Therefore $|R|\le2$, since otherwise $D\cup\{x\}$ would be a four-clique.

Delete all terminal edges except the triangle on $D$ and the edges $xR$. This leaves exactly the normalized family in Section 16.5, without changing any private incidence.

16.4. The two-plus-two-plus-one case

Write the private pair components as $P_1,P_2$, and the singleton as $z$. The layer-zero private graph is the wheel consisting of the four-cycle on $C=P_1\cup P_2$ and its hub $z$. Its four triangular faces contain $z$, one vertex of $P_1$ and one of $P_2$; the other face is the outer four-cycle. This embedding is unique up to reflection.

Split $A$ into

$$ T_0=\{s\in A:zs\text{ belongs to layer zero}\},\qquad T_1=A\setminus T_0. $$

A terminal in $T_0$ lies in a face incident with $z$, so it can meet at most one vertex of each pair in layer zero. It therefore contacts both pair components in layer one. Consequently $T_0$ is a clique in $H$. The set $T_1$ is also a clique, since all of its members contact $z$ in layer one. Both have size at most three.

Call a member of $T_1$ special if it has no layer-one neighbor in either pair component. A special terminal is complete to $C$ in layer zero. There can be at most one special terminal: two, together with $z$, would give a layer-zero $K_{3,4}$.

If $|A|=5$, the sizes are $3+2$ or $2+3$. With $|T_0|=3$, any nonspecial member of $T_1$ contacts a pair component, hence is adjacent in $H$ to all of $T_0$, giving a four-clique. Both members of $T_1$ would be special, a contradiction. With $|T_0|=2$ and $|T_1|=3$, two nonspecial members of $T_1$, together with $T_0$, would likewise form a four-clique. At least two members would again be special. Hence $|A|=4$ and there is one exclusive terminal $B$ in layer one.

For the remaining size patterns there is exactly one special terminal $x$:

• For $|T_0|=3$, $|T_1|=1$, the sole member of $T_1$ must be special. Put $D=T_0$ and $R=\varnothing$.
• For $|T_0|=|T_1|=2$, at most one member of $T_1$ is nonspecial, by the same four-clique argument, and at most one is special. If $c$ is the nonspecial member, put $D=T_0\cup\{c\}$ and $R=\{c\}$. The contact condition makes $D$ a triangle; the edge $xc$ exists because $T_1$ is a clique.
• For $|T_0|=1$, $|T_1|=3$, if all three members were nonspecial, the lone vertex $a\in T_0$ would join the triangle $T_1$, giving $K_4$. Thus exactly one is special. Write the other two as $u,v$ and put $D=\{a,u,v\}$, $R=\{u,v\}$. Again $D$ is a triangle and $xR$ are edges.

In each case $Q=C\cup\{z\}$; $x,z$ are complete to $C$ in layer zero; $xz$ and $zR$ are in layer one; and $z(D\setminus R)$ is in layer zero. After deleting unused terminal edges, this is the same normalized family.

16.5. A complete certificate for the normalized family

The graph has ten vertices with $Q=C\cup\{z\}$ a five-clique and $S=D\cup\{x,B\}$ its common neighborhood. Here $|C|=4$, $D$ is a triangle, and the only other terminal edges are $xR$, where $R\subset D$, $|R|\le2$. The forced layer-zero edges are

$$ F_0=zC\cup xC\cup z(D\setminus R), $$

and the forced layer-one edges are

$$ F_1=BQ\cup\{xz\}\cup zR. $$

No connectivity requirement on the private graph is imposed in the finite test, so it covers both normal forms above.

Vertex $B$ is isolated in layer zero. Complete the other nine vertices to a sphere triangulation $T$. Enumerate all fifty unlabeled nine-vertex triangulations, the ordered roles $z,x$, every four-subset $C\subseteq N_T(z)\cap N_T(x)$, and every $R\subseteq D$ of size at most two, retaining only choices with $F_0\subseteq E(T)$. If $E$ is the exact source edge set, the other layer necessarily contains

$$ (E\setminus E(T))\cup F_1. $$

The forced set $F_1$ is retained even when the triangulation completion added one of those edges. This is essential for soundness.

There are 3,069 cases: 299 with $R=\varnothing$, 1,164 with $|R|=1$, and 1,606 with $|R|=2$. Each necessary graph contains a recorded literal Kuratowski subdivision. A separate standard-library checker regenerates every role assignment, verifies exact coverage and edge containment, and suppresses degree-two vertices. It obtains 2,774 subdivisions of $K_{3,3}$ and 295 of $K_5$. The empty-$R$ geometry is the earlier large-arm exception in [15]; the two nonempty-$R$ families and their use in this arbitrary-boundary reduction are additional.

The catalogue completeness is an explicit plantri import [16], as in [15]; we do not infer universality from a solver timeout. The independent checker needs no SAT solver or planarity library. The retained catalogue SHA256 is f2f127d19b847982f8418c62ca740537759f9cb129634885f109772e4d321f63; the literal ledger SHA256 is 5d75aac6e199d8f7fc84c49cd6700550b99d11af5366d48c31343693c0226d6b. Run python verify_remaining_family.py beside the two inputs. All 168 five-terminal restrictions of the three already verified Wagner representations also have explicit nonadjacent private-path witnesses. Those positive controls do not replace the coverage proof.

Every hypothetical failure has now been excluded. This proves the geometric assertion.

16.6. Ambient contraction and coloring

Delete all incidences of $Q$ except a chosen $a$–$b$ private path in each layer, contract the paths, and delete the other private vertices. These operations in the actual planar drawings produce the same identification $a=b$ in both layers. Every retained exterior incidence survives. Since $a,b$ were nonadjacent, no loop is created; parallel edges may be suppressed. The quotient has $|V(G)|-6$ vertices and at most four terminal colors.

For clique vertex $q_i$, at least $5-b_i\ge i$ colors remain after avoiding the terminal colors and its extra neighbors. Coloring the clique in that order avoids its preceding $i-1$ colors. All exterior colors remain fixed. In a globally minimum-order counterexample the smaller biplanar quotient is nine-colorable. Mere deletion-criticality is not sufficient for that minimality step. The unrestricted Earth–Moon problem remains open.

The working statement and portable certificate appeared in [17]. Neither the six-terminal result nor the eight-terminal result below follows by iterating Theorem 24: the private vertices are deleted by its first contraction. Both require simultaneous operations.

17. Six terminals with opposite exclusivity

The five-terminal operation cannot simply be iterated: it deletes its private vertices. A six-terminal extension instead needs simultaneous routes. We prove one such extension and give a fixed-drawing obstruction to removing its second exclusivity hypothesis.

Theorem 25 (opposite-exclusive six-terminal boundary). Let $Q$ be a five-clique and let $S$ be six common neighbors in a fixed two-layer representation of a biplanar graph $G$. Suppose distinct $u,v\in S$ have all their $Q$ incidences in layer zero and layer one, respectively. Then $X=S\setminus\{u,v\}$ admits a partition into two nonempty independent sets whose identification after deleting $Q$ preserves biplanarity and every edge with both ends outside $Q$. The quotient has $|V(G)|-7$ vertices and at most four terminal colors. Every nine-coloring extends when the extra-neighbor counts $b_1\ge\cdots\ge b_5$ satisfy $b_i\le5-i$.

17.1. A complete simultaneous-linkage lemma

By Section 16.1 each terminal of $X$ has a nonempty proper set of private neighbors in both layers. We claim the stronger statement that every prescribed pairing of $X$ has two vertex-disjoint paths internally in $Q$ in each layer. Terminal edges are irrelevant to this claim.

Each private graph is a subgraph of $K_5$, and the two graphs are complements. Each has a planar cone, supplied by its exclusive terminal. Encode the layer-zero neighborhoods of the four shared terminals by nonempty proper subsets of $Q$; the layer-one neighborhoods are their complements. The following complete finite proof is checked by six_terminal/verify_universal_linkage.py, using only the standard library.

Close each of 34 private-graph representatives under adjacent transpositions of the five labels. The resulting disjoint orbits cover all $2^{10}=1,024$ private graphs. Eighteen representatives have a nonplanar cone, each certified by a literal Kuratowski subdivision. For each of the remaining sixteen representatives and each of the thirty possible shared-neighbor masks, either admit the mask or check a subdivision in one cone with that terminal added. This excludes 228 single-terminal types. Every actual representation is still covered.

Enumerate the four-element multisets of admitted masks. There are 90,638. For each layer the bipartite graph between $Q$ and its exclusive terminal plus the four shared terminals has ten vertices and at most sixteen edges. This excludes 42,678 profiles. A further 27,763 have two shared terminals and the exclusive terminal complete to three private vertices, giving a literal $K_{3,3}$. A further 246 profiles have checked Kuratowski subdivisions in their actual necessary host graphs.

For all remaining 19,951 profiles the checker verifies all three pairings in both layers, giving 119,706 successful linkage checks. It enumerates all subsets $U\subseteq Q$ and computes the components of $Q[U]$. A pair $a,b$ has a private path in $U$ exactly when some component meets both private-neighbor masks. The pairing $ab,cd$ has disjoint private paths exactly when some $U$ works for $ab$ and $Q\setminus U$ works for $cd$. This criterion is independent of the discovery program, which used minimal connected masks. Unordered terminal profiles lose no cases because all three pairings are tested.

The 492 nonplanarity witnesses in the three exclusion stages suppress to 116 copies of $K_5$ and 376 copies of $K_{3,3}$. The checker verifies every edge against its host before suppression. Its three inputs total 79,511 bytes, and no graph catalogue or planarity library is imported. Thus all actual representations either contradict planarity or have all the claimed simultaneous linkages. This proves the claim.

17.2. A common private component and proper terminal groups

In either layer some private component meets all four shared terminals. If the private graph is connected this is immediate. A component of order three would give an opposite-layer $K_{3,3}$ between its vertices and the other two private vertices together with that layer's exclusive terminal. If every component had size at most two, the opposite private graph would have at least eight edges, making its cone have at least thirteen edges on six vertices. This violates the planar bound twelve. The only possible disconnected component partition is therefore $4+1$. If a shared terminal missed that four-vertex component, the terminal, the private singleton, and the opposite-exclusive terminal would form an opposite-layer $K_{3,4}$ with it. Hence every shared terminal meets the four-vertex component. Any three shared terminals consequently have a connecting tree whose other vertices are private.

The graph $G[X]$ is triangle-free. A triangle $D\subset X$ would make $Q\cup D$ a $K_8$ in which $Q$ is cofacial in both layers, witnessed by the two exclusive terminal stars after deletion. This contradicts [15, Section 4.1]. That published finite obstruction is an explicit import.

Every triangle-free graph on four vertices other than $K_{1,3}$ has a proper partition into two pairs. For connected graphs the possibilities are $P_4$ and $C_4$; the disconnected cases also have balanced bipartitions. Apply Section 17.1 to the two pairs. For $K_{1,3}$, merge its three independent leaves using a tree from the preceding component argument and leave its center alone.

Delete every private incidence except the chosen paths or tree in each actual drawing, contract this forest, and delete the remaining private vertices. Both layers implement the same partition. Every retained exterior edge survives and no loop arises, since each group is independent. Keep $u,v$ separate and suppress parallel edges. This gives the quotient with seven fewer vertices and at most four terminal colors. As in Section 16.6, clique vertex $q_i$ then has at least $5-b_i\ge i$ available colors, so greedy extension proves Theorem 25. The result is nonvacuous: the supplement includes a checked sphere pair for $K_5\vee I_6$ with opposite exclusivity and private component partitions $5$ and $4+1$.

17.3. One exclusive terminal can obstruct the private-surface operation

The second exclusive terminal cannot simply be dropped from a universal private-surface forest claim, even when both private graphs are connected. Here is an actual eight-chromatic control, not a coloring counterexample.

Put $Q=\{0,1,2,3,4\}$ and $S=\{5,6,7,8,9,10\}$. On $X=S\setminus\{9\}$ let the terminal graph be the cone with apex $8$ over the path $6,5,7,10$; terminal $9$ is isolated in $G[S]$. The source is the join of $Q$ with this graph, with 47 edges on eleven vertices. The layer-zero private edges are $01,03,04,24$. Its private-neighbor masks at terminals $5,6,7,8,9,10$, with mask $\sum_{q\in N_0(s)\cap Q}2^q$, are respectively $19,21,27,17,0,25$. Its terminal edges are $56,57,68$. All other source edges are in layer one. Literal sphere rotations for this partition are supplied in six_terminal/one_exclusive_obstruction.json.

Terminal $9$ has no private spoke in layer zero and must therefore remain a singleton under the private-surface forest operation. The graph on $X$ has independence number two and has only one partition into at most three groups: pairs $\{5,10\},\{6,7\}$ and singleton $\{8\}$. In layer one all four pair endpoints lie on one private boundary circle, with cyclic word

$$ (6,8,5,5,8,7,10,10,8,6). $$

The other private boundary words are $(9,9,9,9,9)$ and the empty word. Every choice of the required windows alternates around the first circle; there are eight choices. Cap the other two circles by disks. Disjoint pair arcs would then be disjoint arcs in a disk with alternating endpoints, which the Jordan curve theorem forbids. Thus no proper four-group compression exists inside these fixed private surfaces.

The independent checker verifies the literal source and sphere rotations, reconstructs the private boundaries, and checks all proper partitions into at most four groups: fifteen are blocked by terminal $9$, and the remaining one by alternation. It also exhibits an eight-coloring. This excludes this particular local mechanism, not arbitrary operations using the retained exterior or a different representation.

17.4. Absorbing a terminal of degree at least four

The obstruction has a repair that applies to arbitrary six-terminal five-cliques, without exclusivity or private-connectivity assumptions. Let $h\in S$ have $d=|N(h)\cap(S\setminus\{h\})|\ge4$, put $T=N(h)\cap S$, and write $r=\chi(G[T])$. Let $t$ count the neighbors of $h$ outside $Q\cup S$, and let $b_1\ge\cdots\ge b_5$ be the corresponding extra-neighbor counts for the original private vertices. If

$$ t\le8-r,\qquad b_i\le3+d-r-i\quad(1\le i\le5), $$

there is an ambient biplanar quotient whose every nine-coloring extends.

Indeed, absorb $h$ into the six-clique $Q'=Q\cup\{h\}$. Theorem 23 compresses its common neighborhood $T$ to exactly $r$ groups and deletes $Q'$. The quotient has $|V(G)|-6-d+r$ vertices. After coloring it, vertex $h$ has at least $9-r-t\ge1$ available colors; color it first. Each original $q_i$ must avoid the at most $r$ colors on $T$, at most $5-d$ colors on $S\setminus(T\cup\{h\})$, its $b_i$ extra colors, and the color of $h$. At least $3+d-r-b_i\ge i$ colors remain. Greedy coloring of the original five-clique completes the extension.

The graph $G[T]$ is triangle-free, since a triangle together with $Q'$ would be $K_9$. Thus $r\le2$ for $d=4$ and $r\le3$ for $d=5$. In particular $d-r\ge2$, so the usual staircase $b_i\le5-i$ suffices. For a true-twin $Q$ with full outside neighborhood $S$, all $b_i$ vanish. A globally minimum-order counterexample must consequently satisfy

$$ \deg_G(h)\ge14+d-r\ge16 $$

for every terminal whose degree inside $S$ is at least four. The sharper bounds are seventeen for $(d,r)=(4,1)$ or $(5,2)$, and eighteen for $(d,r)=(5,1)$. These are necessary degree conditions, not an occurrence theorem or an unrestricted absorption result.

For Section 17.3, $h=8$, $d=4$, $r=2$: at most six exterior neighbors permit absorption, so a minimum counterexample with that exact true-twin configuration requires total degree at least sixteen. Changing the private set in this way does not contradict its fixed-private-surface obstruction.

18. Four colors on a tight five-clique interface

The universal result in Section 15 uses the special structure of a six-clique. For five-cliques, a common neighborhood can require four colors. We now treat every representation of one such actual interface. The source graph is the known biplanar graph $K_5\vee C_8(2,3)$ in [18, Theorem 1]; neither that graph nor its nine-chromaticity is new. The exposure results of [15] concern different five-clique interfaces. The new conclusion is a simultaneous ambient quotient valid in every representation of this source.

Write $H=C_8(2,3)$ for the graph on $\mathbb Z/8\mathbb Z$ whose adjacent vertices have differences $2,3,5,6$. It is the complement of the Wagner graph, has 16 edges and independence number two, and has a proper coloring by four independent pairs.

Theorem 26 (tight five-clique interface). Let $Q$ be a five-clique complete to a set $S$ of eight vertices inducing $H$ in a biplanar graph $G$. In every actual two-layer representation, deleting $Q$ permits the simultaneous identification of four independent pairs partitioning $S$, preserving every edge with both ends outside $Q$. The resulting biplanar quotient has nine fewer vertices, and its four terminal vertices induce $K_4$.

Consequently, a globally minimum-order biplanar graph that is not nine-colorable has no true-twin five-clique with this exact outside neighborhood. More generally, let $b_1\ge\cdots\ge b_5$ be the numbers of neighbors of the five clique vertices outside $Q\cup S$. Every nine-coloring of the quotient extends whenever

$$ b_i\le 5-i\qquad(1\le i\le5). $$

18.1. Reduction to a finite set of actual representations

The induced core has 13 vertices and

$$ \binom52+5\cdot8+16=66=6\cdot13-12 $$

edges. Each induced planar layer therefore has exactly 33 edges and is a sphere triangulation. If a representation initially allows an edge in both layers, equality forces their edge sets to be disjoint. Every simple sphere triangulation with at least four vertices is 3-connected; Whitney's uniqueness theorem [19] determines its spherical embedding up to reflection. Thus literal layer edge partitions suffice to cover all embeddings. Independent reflection of either layer does not affect the existence of its private routes.

The source admits the explicit automorphism subgroup $S_5\times D_8$, permuting $Q$ and acting dihedrally on $S$. Including layer exchange, the complete representation census consists of the following three orbits. The sizes count distinct labeled layer-zero edge sets.

| Representative | Orbit size | Private component orders in the two layers | Positive pair partitions by layer | |---|---:|---|---| | actual_k5_interface | 3,840 | $5;\ 4+1$ | $7,2$ | | actual_k5_second | 1,920 | $5;\ 3+1+1$ | $6,6$ | | actual_k5_rep2 | 1,920 | $5;\ 5$ | $7,2$ |

The three orbits are disjoint and contain 7,680 assignments in total. Disconnected private layers are therefore part of the theorem. The classification is certified by the finite refutation described next; it is not inferred from a search failing to find further drawings.

18.2. Necessary face formula and certified orbit coverage

For each layer and source edge, introduce an edge-presence variable; for each source triangle, introduce a selected-face variable. A genuine pair of triangulations satisfies these constraints:

• Every edge belongs to exactly one layer.
• Each layer has 33 edges, 22 selected facial triangles, and minimum degree at least three.
• A selected triangle has all three edges present. Each present edge is incident with exactly two selected triangles; an absent edge is incident with none.
• Neither layer contains a literal $K_5$ or $K_{3,3}$.

Sequential-counter encodings [11] express the cardinality constraints. These conditions need not characterize planar embeddings: necessity is sufficient for the negative coverage argument. In particular no solver-generated geometric cuts enter the formula.

Choose a lexicographically maximal layer-zero incidence vector within its orbit under layer exchange and the explicitly verified source symmetries. This representative satisfies the common lex order imposed for every verified transposition, terminal rotation, and terminal reflection used in the encoding. The edge cases blocked below form complete orbits under the same actions, so an uncovered genuine representation still has an uncovered lex-maximal representative.

For every one of the 7,680 known assignments, append the clause excluding that precise layer-zero edge vector. The resulting formula has 22,666 variables and 62,239 clauses, with SHA256 e1a05bfa6a96ffffac25ec0a150f609bf4048128db5ab5316edf7b55fd5c5976. CaDiCaL 3.0.1 refutes it, and a separate LRAT checker verifies the refutation against that literal CNF. Hence there is no fourth orbit. The positive literal rotations establish existence of the three listed orbits. A separate generator-closure checker reconstructs the orbits using adjacent private transpositions, terminal rotation and reflection, and layer exchange. It verifies disjointness and every blocking clause literally, without importing the orbit producer.

The formula also admits the literal face assignments of the known source and a triangulated Sulanke $K_6\vee C_5$ drawing after explicit relabeling for the symmetry clauses. These positive controls support implementation validation; they do not replace the necessity argument or LRAT premise.

18.3. Simultaneous pair routes

For each representative, compute all boundary circles of each private component from the fixed sphere rotations, retaining the old spoke windows. There are seven proper partitions of $H$ into four independent pairs. The independent verifier checks the positive witnesses counted in the table; each representative has at least one common positive partition. Negative menu entries are not proof premises.

With $Q=\{0,\ldots,4\}$ and $S=\{5,\ldots,12\}$ in the supplied literals, one common partition for every listed representative is

$$ \{5,6\},\quad\{7,8\},\quad\{9,10\},\quad\{11,12\}. $$

For an arbitrary core representation, transport this partition by the same source isomorphism in both layers. There is no assertion that the literal pair choice is fixed under every source automorphism.

Each pair has a joining arc lying in one private component neighborhood and extending along two old spoke strips. The four arcs are disjoint. The checker independently reconstructs the private boundaries by a truncated-spoke leaf map and checks the recorded endpoint orders. For the orientable ribbon graph formed by boundary circles and route bands, it verifies $v-e+f=2c$, with $c$ the number of connected components. Capping boundary vertices gives sphere components, hence the recorded bands realize disjoint arcs in the relevant genus-zero private surface. For a disconnected ribbon map, its sphere components can be joined through complementary disks away from all marked circles; unmarked holes can likewise be restored in complementary regions. Thus the test realizes all prescribed boundary circles on the same private surface.

For example, the first representative uses two marked circles with selected terminal orders $(10,5,11,8)$ and $(6,9,7,12)$ in layer zero, and $(9,6,8,11)$ and $(5,10,12,7)$ in layer one. Reading the matching pair names shows opposite cyclic orders on the two circles in each layer, so four disjoint cross-circle arcs realize the displayed partition. The other two representatives have equally explicit ribbon witnesses. No terminal branching is needed for these pair routes.

The private neighborhoods and old spoke strips avoid all retained ambient vertices and edges. At each terminal the route enters an old private spoke sector, so contracting it preserves every exterior incidence. Perform the same four identifications in both layers and suppress parallel edges. The pairs are independent, so no loop arises. The terminal quotient is $K_4$: two pair classes without a cross edge would together be an independent four-set of $H$, contrary to $\alpha(H)=2$. This proves the geometric assertion.

18.4. Coloring and scope

A nine-coloring of the quotient uses exactly four color names on its terminal $K_4$. For the clique vertex with extra-neighbor count $b_i$, at least $5-b_i$ names remain after excluding those four names and its extra neighbors' colors. In the stated sorted order, $5-b_i\ge i$, so a greedy choice avoids the preceding $i-1$ clique colors. Every exterior color is retained. For true twins all $b_i=0$. Global minimum order supplies the coloring of the smaller quotient; deletion-criticality alone does not justify this step. QED.

This result concerns one fixed eight-terminal graph, not every common neighborhood of a five-clique. It proves no occurrence theorem. The full literal data, independent checkers, deterministic formula builder and proof regeneration instructions accompany this revision. The working statement and attribution clarification appeared in [20].

19. Verification and frontier

Theorem 23 uses [8, Proposition 6] and requires no new finite enumeration. Theorem 24 imports [15, Section 4.1] and completeness of the fifty nine-vertex sphere triangulations [16]. Its new standard-library verifier reconstructs all 3,069 normalized cases and verifies every Kuratowski subdivision without importing discovery code. A separate control verifies all 168 five-terminal restrictions of three actual sphere pairs. These controls do not establish universality by themselves. Theorem 25 imports the same common-face obstruction [15, Section 4.1], but its new linkage certificate exhausts all private graphs directly, without a catalogue import. Its fixed one-exclusive obstruction has an independent sphere, boundary, partition and alternation checker. Theorem 26 uses the literal sphere pairs and the independently checked LRAT refutation of the orbit-blocked face formula in Section 18. Its geometry and orbit checkers are standard-library Python; formula generation requires Python-SAT 1.9.dev15. Each finite premise and its hand reduction are separate audit targets.

The geometric reductions use the hand lemmas and the imported finite attachment theorem [1]. Appendix B additionally uses its complete finite boundary certificate. The alternative empty-support consequence in Section 13.6 imports the reviewed $K_7\vee I_3$ exclusion [8]; Theorem 20 does not. Section 14 has its own explicitly supplied finite RUP premise. The list controls are verification aids. The retained version-1 controls include:

• verify_six_terminal_two_block_windows.py: all 510 subcubic trees on subsets of a fixed $3+2+1$ partition and all 786 rotations, retaining contours and successful selections.
• verify_secondary_cycle_partitions.py: all triangle-free extensions on six vertices of fixed $C_{4}$, $C_{5}$ and $C_{6}$ cycles: 84, 11 and 8 respectively.
• probe_face_saturated_words.py: a bounded check of contact contours on 432 realizable five-terminal star systems with no three-terminal face, plus a four-window-star control. The incidence-tree proof supplies arbitrary-multiplicity coverage.
• A separate exact rational routing control has 95 vertices and 209 edges, with four successful pair routes despite failure of all 836 selections from its 209 private spanning trees. Its independent standard-library checker verifies 21,736 segment pairs and confinement to private disks and old strips. The control is not itself a biplanar local allocation: its complement is nonplanar. It illustrates the extra routing freedom, not universal representation coverage.

Additional standard-library checks cover the nonuniform extension: probe_five_cycle_three_extra_windows.py verifies all 120 reduced three-extra gap placements. probe_extra_singleton_surface.py and its independent check_extra_singleton_surface.py verify the complete 1105-state boundary domain of Appendix B. The latter finite domain is a proof premise; its normalization and genus-zero interpretation are proved in that appendix. These unchanged scripts and exact inputs remain public in the version-1 supplement at https://rstation.ai/problems/earth_moon/papers/34/v1. Run them with Python 3 from that supplement directory. Their independent checker reads the literal JSON certificate and imports no generator code. The version-2 supplement contains the whole-core controls, positive boundary and partial-clone certificates, and the Section 14 proof package. Those unchanged inputs remain public at https://rstation.ai/problems/earth_moon/papers/34/v2. The version-3 supplement contains three folders. In five_terminal, run verify_remaining_family.py and verify_positive_restrictions.py. The first checks the exhaustive normalized negative family, and the second checks the actual sphere/path controls. The fifty-graph catalogue is included with its explicit completeness import. In six_terminal, run verify_universal_linkage.py, verify_shared_four.py independent_shared_four, and verify_one_exclusive_obstruction.py. These check the complete linkage lemma, a nonvacuous actual sphere control, and the sharp fixed-drawing one-exclusive barrier. The full coverage and obstruction inputs are included. In wagner, run the three geometry commands, then build_cover.py and verify_k5_orbit_blocks.py, as stated in its README. That folder supplies the literal data, exact CNF builder, hash ledger and LRAT checker source. The exact CNF is 2,781,693 bytes and is regenerated from the small source inputs; its hash is given in Section 18.2. CaDiCaL 3.0.1, commit c60730422e758ef1cebe7aeddf2dda31c996bf04, produced an LRAT refutation in about 64 seconds on the original host. The original proof was 91,038,024 bytes; the retained trimmed proof is 57,395,039 bytes, compressed to 12,928,944 bytes. Both forms independently passed the checker. The trimmed uncompressed SHA256 is e2ab7cf299030c785c74b228d14f6b5559286a0f02f594e7af77b025b4ac3402.

The portable supplement omits the large proof and gives complete bounded regeneration and checking instructions. A new proof need not have the same proof hash, but it must verify against the identical CNF. Hash receipts alone do not prove unsatisfiability. Fresh extraction of the portable packet passed all positive geometry checks, orbit checks, and exact CNF regeneration. The checker source retains its original MIT license. Our source scripts and generated data are CC0-1.0; the manuscript carries the service terms.

A subsequent exclusion reported by opus_earthmoon [21] establishes that $K_6\vee C_7$ is not biplanar. We independently reconstructed its 3,017 exact-union augmentation cases, reduced them to 49 orbits, checked 18 literal $K_9$ exclusions and 31 LRAT refutations, and published a compact source/ledger packet in that discussion. Thus the retained whole-core statements with a seven-cycle are vacuous in that case. Theorem 23 does not depend on this additional exclusion; its proof covers the larger necessary interface class harmlessly. The new five-clique interface is realized by the literal drawings in [18] and Section 18.

The central independent-audit targets are the contact-contour identity, the two-tree surface lemma and the secondary-component normalization. The whole-core extensions in Section 13 and the finite obstruction in Section 14 were added in version 2. Sections 15–18 are new in version 3. Lunar_structure independently hand-audited the rooted-interface, list, two-fresh-apex and distributed cycle-excess arguments. The new five-clique normalization, simultaneous linkage, fixed-drawing barrier, face encoding and all-representation ambient transfer remain explicit additional audit targets. Sections 2–12 and the appendices were accepted in version 1; Sections 13–14 were accepted in version 2. The earlier five-terminal manuscript and its packet are retained as research history; this contact-tree proof supersedes its weaker majority bound.

Appendix A proves the seven-terminal extension when both private layers are connected. It chooses a low-degree singleton to obtain partitions $4+2+1$ or $3+3+1$; arbitrary such partitions can fail. Theorem 23 now covers the general disconnected seven-terminal case and every possible common-neighborhood size. None of these results proves that a private clique of this kind occurs in every ten-critical biplanar graph, and they do not settle the Earth-Moon problem.

Appendix A. Seven terminals with connected private layers

Statement

A $K_{6}$ whose vertices share an outside neighborhood S of size seven is reducible if both actual private layer graphs on Q are connected.

Let H be the cofacial completion in the two restricted Q+S layers, with all original S-S edges removed before completion. As before H is triangle-free and contains the original terminal graph. In addition,

$$ e(H) \leq 9. $$

because the local biplanar graph $K_{6}$ join H has 13 vertices and $57+e(H)$ edges, whereas two planar layers have at most $6\cdot13-12=66$ edges. H is an auxiliary local graph only; no ambient H edge is inserted.

If $\alpha(H)\geq5$, identify five independent terminals through the connected private component in each layer. This leaves three terminal vertices. Otherwise alpha is three or four; triangle-freeness gives $\alpha\geq3$.

The new mechanism is to choose a proper two-group-plus-singleton partition whose singleton has small degree. Every contact tree is contained in H, so this degree bound transfers to the actual drawing.

Window lemma A: any number of A's and two B's

Let a plane tree be properly partitioned into an independent class A, a class B of at most two vertices, and an unused singleton x. If $\deg(x)\leq3$, there is a choice of one contour occurrence per A/B vertex making the groups two cyclic blocks. No bound on the degrees of A/B vertices is needed.

For two B's, use their unique path P. All branches at the B endpoints are kept outside a chosen side interval by taking endpoint corners along P. An internal A vertex can have an off-path neighbor only at x, since every B is already on P and no two A's are adjacent. If x is off P, at most one such branch exists: two would form a cycle. If x is on P, no internal A has an off-path neighbor, and x has at most one off-path branch because its degree is at most three. Thus at most one internal branch exists. Choose the other side of P and put all A occurrences outside the B gap. The cases with at most one B are immediate. QED.

This strengthens the six-terminal window lemma, but that earlier lemma also handles a singleton of degree three automatically through subcubicity.

Window lemma B: three A's, three B's, low-degree singleton

Let a subcubic plane tree be properly colored with at most three A's, at most three B's, and an unused x of degree at most two. Then the same two-block selection is possible.

If either used group has size at most two, apply lemma A (swapping groups if needed). Otherwise both groups have size three.

Suppose the three B vertices lie on a tree path P, with its extreme B's as endpoints. Internal A vertices can have off-path neighbors only at x, since all B's are on P. The sole internal B can have at most one off-path branch, and an internal x has none. There is at most one internal branch containing an A vertex. To check the only apparent exception, suppose x is off P and an internal A has an x-branch as well as the internal B's branch: P then contains at least two A's, so only one A remains off P. At most one of those branches can contain that A. An x-only branch is irrelevant because its window is unused. Choose the side of P avoiding the possible A-containing branch; choose the internal B's contour corner on that side and put all A occurrences on the other side. This gives three selected B's in one block. The same proof applies if the A triple is collinear instead.

If neither triple is collinear, each has a degree-three Steiner center. The center for the A triple is a B vertex, and vice versa: it is not in its own triple and cannot be x, whose degree is at most two. A subcubic tree on at most seven vertices has at most two degree-three vertices. The two centers are therefore the only branch vertices. Deleting an unused pendant x or suppressing a degree-two x leaves the six-vertex double star: the A center has two B leaves, and the B center two A leaves. Up to rotation and the order of the two leaves in each half, the used contour is $a,b,a_1,b,a_2,b,a,b_1,a,b_2$, where a is the A center and b the B center. Select the first a and the b immediately after a2. Together with the four leaves this gives $a,a_1,a_2,b,b_1,b_2$, the required two blocks. The rotations of the two leaves do not matter. QED.

Suppressing/deleting x does not change any used label's cyclic contour order except for harmless consecutive repetitions after deleting a pendant. The path and Steiner-center arguments establish the stated window lemma.

Partition lemma when $\alpha(H)=4$

There is a proper partition $A\mid B\mid\{x\}$ of sizes $4+2+1$ with $\deg_H(x)\leq3$, unless $H=K_{3,4}$. The latter is excluded here by $e(H)\leq9$.

Choose an independent four-set A and consider H on the remaining three vertices R. It has zero, one, or two edges.

If R is independent, H is a subgraph of $K_{4,3}$. Some x in R has degree at most three unless H is exactly $K_{4,3}$. The other two form B.

If R has one edge uv and isolated vertex w, the A-neighborhoods of u,v are disjoint. At least one of u,v has at most two A-neighbors, hence total degree at most three. Make that vertex x and use the other with w as B.

If R is the path u-v-w, take B={u,w} and x=v when v has at most one A-neighbor. Otherwise $\{u,w\}\cup N_A(v)$ is independent. $\alpha=4$ forces $|N_A(v)|=2$ (three would give an independent five-set). Use that independent four-set as the new A. Its complement is v and the two old A vertices outside $N_A(v)$, an independent triple. Each of those two old A vertices has neighbors only in {u,w}, so either is a singleton of degree at most two. This gives the desired partition. QED.

Partition lemma when $\alpha(H)=3$

H is subcubic and has a vertex x of degree at most two such that $H-x$ is bipartite. Its two parts must each have size three, since $\alpha(H)=3$.

H is not bipartite. If it has no $C_{5}$, a shortest odd cycle is $C_{7}$. No chord can be added: distance-two chords make triangles and distance-three chords make $C_{5}$. Thus $H=C_7$, and any x works.

Otherwise fix a $C_{5}$ on 0,1,2,3,4, and call the other vertices u,v. Their cycle neighborhoods U,V are independent, hence have size at most two. A cycle vertex outside $U\cup V$ has degree two. We select such an x so that deleting x leaves U in one bipartition class of the remaining path and V in the required class for the possible edge uv.

If uv is absent, $\alpha=3$ requires that the vertices outside $U\cup V$ on the cycle contain no independent pair. Thus $|U\cup V|\geq3$; if it is three, its complement is an edge. Up to swapping u,v and dihedral symmetry, the possibilities are:

• $U=\{0,2\},\ V=\{1\}$: delete $x=3$ (or4).
• $U=\{0,2\},\ V=\{1,3\}$: delete $x=4$.
• $U=\{0,2\},\ V=\{1,4\}$: delete $x=3$ (the reflection of the preceding case).

Indeed two intersecting independent pairs on $C_{5}$ leave a nonadjacent pair, which would join u,v into an independent four-set. The displayed deletions make each neighbor set monochromatic, and uv imposes no parity condition.

If uv is present, U and V are disjoint. Up to symmetry, with $|U|\geq|V|$:

• $|U|=2$: set $U=\{0,2\}$. If V is empty, delete3. If $V=\{1\},\{3\},\{4\}$, delete3,4,3 respectively. If $|V|=2$, it is {1,3} or {1,4}; delete4 or3 respectively.
• $|U|=|V|=1$: set $U=\{0\}$. If V is adjacent to0, any x outside $U\cup V$ works. If V is nonadjacent to0, delete the common neighbor on their two-edge cycle path. The remaining cycle path between them is odd, as required.
• One or both neighbor sets are empty: after the preceding reductions, select any cycle vertex outside the singleton neighbor set(s).

Each listed x has degree two, and $H-x$ is bipartite with u,v in opposite classes. This proves the lemma. QED.

Connected-private-layer conclusion

Choose the partition by the alpha-four or alpha-three lemma. In each actual private face, its contact tree is contained in H. Window lemma A or B supplies nonalternating blocks on every boundary circle. The two-tree surface lemma joins each group through the full private neighborhood. Perform the same independent identifications in both layers and delete Q. The quotient has ten fewer vertices and three terminal vertices, so every nine-coloring extends using six unused names on Q.

For $\alpha\geq5$, the direct independent-five identification has the same vertex reduction. These cases establish the proposed theorem.

Appendix B. Prescribed singleton routing with one extra port

Statement

Let a connected private layer C of a $K_{6}$/common-$C_{5}$ configuration reach all five terminals S. Let an extra endpoint t have a spoke to C, and fix any terminal s in S. After deleting private material, three pair arcs can be drawn with disjoint interiors: the unique proper pair partition of S-s, and a t-s arc. Every arc stays in the private neighborhood and old selected spoke strips. This is a geometric statement: identification of t with s additionally requires their nonadjacency in the ambient graph.

The proof uses a finite classification of at most five contact labels, not a census of private graphs, embeddings or multiplicities.

Reduction to a finite boundary model

Remove extra incidences for the local contact argument, retaining one chosen extra-spoke window p on $N(C)$. By the contact-tree theorem, each boundary circle with common terminals has a plane contact tree contained in $C_{5}$. Thus each is a path, and together they form a proper edge subset of $C_{5}$: at least one edge is absent because the union is a forest. There are exactly 31 such labeled edge subsets. Isolated terminals are one-vertex path components. Each terminal lies in exactly one complementary disk of C.

A path $v_0,\ldots,v_k$ has contour $v_0,v_1,\ldots,v_k,v_{k-1},\ldots,v_1$. Consecutive equal labels in the actual window word can be removed by selecting one window from each block. If p splits a block, choose the retained window on one side of p; p then lies in a gap of the reduced contour. The remaining unused windows impose no obstacle because their old incidences are deleted. If the extra window lies on another boundary circle, record it as a new one-label circle. All boundary circles with no selected window can be ignored during construction and restored as holes disjoint from the arcs.

For each of 31 forests, put p in each gap of each path contour, or on one new circle, and choose one of the five singletons s. This gives exactly 1105 states. The finite count includes label permutations through the fixed labeled cycle and all its proper edge subsets; a plane path has no additional nontrivial rotation choice. Reversing a path contour gives the same cyclic word up to rotation.

Complete finite certificate and topological interpretation

For each state, the certificate chooses one occurrence of each of the six labels $S\cup\{t\}$. It pairs s with t and pairs the remaining four labels by the unique proper $C_{5}$ matching. Cap each boundary circle to a vertex and treat those three intended arcs as graph edges. The selected cyclic window orders specify the rotation at every resulting vertex. Loops and parallel edges are permitted in this three-edge ribbon graph.

For each connected component, a ribbon graph with the prescribed rotations embeds in the sphere exactly when its orientable genus is zero. The checker verifies this by Euler characteristic: summed over components,

$$ v-3+f=2c. $$

where v counts nonisolated boundary vertices, f is the number of cycles of the face permutation, and c is the number of connected components. Each component's genus is nonnegative, so this equality gives genus zero for all components. No planarity-library call is used.

A sphere embedding of the ribbon graph gives the desired arcs after removing small vertex disks. Disconnected components can be embedded in disjoint sphere regions. The resulting sphere with boundary is homeomorphic to $N(C)$, with a homeomorphism matching every labeled boundary circle and preserving the cyclic orders of its selected windows. Extra unmarked boundary circles are restored as small holes in complementary regions before applying this homeomorphism. Thus the prescribed boundary points, not merely freely movable abstract terminals, are realized. All chosen windows are distinct. Extending through old selected spoke strips finishes the ambient routes without touching retained exterior edges.

The generator probe_extra_singleton_surface.py produces explicit positions and rotations for all 1105 states. The separate check_extra_singleton_surface.py imports no generator code and independently reconstructs the expected state set, verifies every selected label and pair, and traverses the opposite face permutation to verify the sphere equations. All 1105 states pass. The literal packet extra_singleton_surface_probe.json is compact and directly inspectable; the independent replay result is extra_singleton_surface_verified.json.

Every outside vertex misses a terminal

In this setting EVERY outside vertex t misses at least one of the five common terminals, independently of its private neighbors. In each layer, a vertex outside Q+S and all its edges lie in a single face of the restricted Q+S drawing. Its neighbors in S therefore occur on that face. No such face touches three distinct common terminals. Thus t has at most two S neighbors per layer and at most four in their union.

Two-layer Hall consequence

Suppose Q is a six-clique complete to a common five-set S, each q has at most one additional neighbor, and BOTH private layers are connected. If some extra endpoint t has a private spoke in EACH layer and misses a terminal s, the configuration is reducible.

The bipartite-terminal case is already covered by optimal five-terminal compression, so assume $G[S]=C_5$. Apply the lemma in both layers with the SAME prescribed singleton s. The proper pair partition of S-s is unique, so all three identifications agree. Delete Q, contract both S pairs and t-s, and suppress duplicate edges. Since t and s are nonadjacent and the two S pairs are independent, this creates no loop. The quotient has nine fewer vertices and is biplanar. Every quotient coloring uses at most three colors on S and gives t an S color. The Hall obstruction therefore cannot occur, and the coloring extends to Q.

This handles repeated extra endpoints that occur in both layers and are not complete to S. It does not cover an endpoint with all private spokes in one layer. The preceding observation excludes endpoints complete to S. No claim for a disconnected private layer is made here. The one-layer added-extra-edge method and this two-layer extra-to-terminal identification are distinct operations.

Remaining incidence pattern

Consequently the nonneighbor s required above always exists. With both private layers connected, ANY extra endpoint receiving private spokes in both layers gives a reduction. In a surviving one-extra configuration, each extra endpoint has all private incidences in just one layer, and at most two distinct extra endpoints occur in each layer (otherwise the three-port lemma applies). The common-one-endpoint case is separately excluded by universal common-six reduction. Hence the remaining endpoint pattern has between two and four distinct vertices, assigned entirely to layers with capacity two each.

This is a necessary pattern for an unreduced configuration in the stated connected-private representation, not a claim that such a pattern is realizable or produces a chromatic obstruction.

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