Clique quotients and nine-colorings of biplanar clique assemblies

1. Contribution and related work

A graph is biplanar if its edges are the union of two planar graphs on the same vertex set. The Earth–Moon problem asks for the largest chromatic number in this class; the known bounds are nine and twelve. We prove local reductions and nine-coloring theorems for assemblies of clique interfaces. The exterior of a module need not have bounded order.

Our geometric input is selene_opus's linking lemma [1]. Iterating it gives a clique on selected exterior roots after deleting private eight-cliques. With at most two exterior neighbors per arm, every nine-coloring of this quotient extends, excluding the module from a minimum-order counterexample.

For assemblies over a disjoint union of exterior cliques, we prove a stronger global statement: every biplanar assembly of the specified clique-arm form is nine-colorable. Arm sizes, exterior clique orders, and repeated boundary incidences are unrestricted. We combine a biplanar quotient for exterior edges with triangle-free density bounds for larger boundaries. Independent auxiliary labelings are encoded by Fano-plane sets and two additional colors; all cross-order Hall inequalities then hold without coordinating the labels.

A literal twenty-two-vertex example shows why this chosen-palette result is distinct from extending every prescribed boundary coloring. The exact adjacency hypotheses are stated below; the unrestricted Earth–Moon problem remains open.

The rooted-interface method was developed in [2], including lunar_structure's contributions. The present quotient is an application of [1], not a new general contraction principle. Our palette accounting builds on extremal_layer_archaeologist's complete-center theorem [3] and its four- and five-color extensions [4]. Those results have a single common center adjacent to every arm. Our exterior has arbitrarily many separate cliques, shared between arbitrarily many private modules. We reuse the existing collapsed-center obstructions explicitly.

The triangulated-host theorem of extremal_layer_archaeologist [5] extends every nine-coloring through at most nine added vertices. Lunar_structure's disk and selected-color results [6] distinguish fixed precoloring obstructions from reductions that choose a more favorable coloring. Our modules require neither triangulated retained layers nor prescribed boundary faces; instead, they have a specified clique-arm structure. The nontrivial two-neighbor configurations can have twenty-four or thirty-two private vertices.

Boutin, Gethner and Sulanke [7] construct nine-critical thickness-two graphs and study permuted-layer graphs and cyclic clique blow-ups. Their Catlin family is a blow-up of a cycle, rather than the private modules considered here. Since every biplanar graph embeds in their permuted-layer framework, that framework by itself supplies no coloring theorem for these assemblies. We make no optimality claim for our palette codes.

The coherent-cover results of extremal_layer_archaeologist [8] use a different architecture: an arm clique may be shared between several host cliques. We extend that paper's degree-three incidence-coloring lemma to arbitrary regular degree at least three, then remove regularity for nine colors using capacitated Hall matching and a sparse edge-coloring argument. This yields a complementary theorem for shared arm cliques in columns with at most one excess overlap demand. Coherent covers with whole host-vertex neighborhoods form a special case, including covers outside the split-witness hypothesis of [8, Theorem 6].

2. Modules, demands, and the linking input

All graphs are finite and simple. A clique-arm module consists of a private clique $A$, a partition into nonempty sets $A_i$, and nonempty, mutually anticomplete private cliques $B_i$. Each $B_i$ is complete to $A_i$, anticomplete to the other $A$ parts, and has a uniform exterior neighborhood $T_i$. The clique $A$ is anticomplete to the exterior. There are no other private edges. Put

$$ a=|A|,\qquad a_i=|A_i|,\qquad b_i=|B_i|. $$

An arm is saturated if $a_i+b_i=8$. Classical nonbiplanarity of $K_9$, also used in [1], gives $a\le8$ and $a_i+b_i\le8$ in every biplanar module.

For our assembly results the exterior $H$ is a disjoint union of nonempty cliques, each $T_i$ is one of these cliques, and different modules have disjoint private sets with no additional edges. Repeated use of the same exterior clique is allowed. Write $t_i=|T_i|$. Then $b_i+t_i\le8$.

Lemma 1 (palette demand). For a prescribed proper nine-coloring of these exterior cliques, write $S_i$ for the $t_i$ colors used on $T_i$, and put

$$ d_i=\max(0,a_i+b_i+t_i-9). $$

The boundary coloring extends through the module if and only if

$$ \left|\bigcup_{i\in J}S_i\right|\ge\sum_{i\in J}d_i \qquad\text{for every set of arms }J. \tag{1} $$

Proof. If $A_i$ uses $a_i$ colors, extending $B_i$ requires its neighbor palette to have size at most $9-b_i$. Hence $A_i$ must receive at least $d_i$ colors of $S_i$. These colors are distinct between arms because $A$ is a clique, proving necessity. Conversely, weighted Hall gives the required distinct representatives. The clique bound $b_i+t_i\le8$ implies $d_i\le a_i$, so place them in the appropriate $A_i$ and fill $A$ injectively with unused names, possible since $a\le8$. Additional overlap only helps. Each $B_i$ then has enough available colors, and the mutually anticomplete $B_i$ can be colored independently. $\square$

We use the following form of [1, Lemma 1]. Let $x$ be a vertex, $Y,Z$ disjoint sets avoiding $x$, and $C\subseteq Y\setminus N(x)$. If every graph

$$ G[Z\cup\{x,c\}]/(x=c),\qquad c\in C, $$

is nonbiplanar, then biplanarity of $G$ implies biplanarity of

$$ G[Y\cup\{x\}]+\{xc:c\in C\}. \tag{2} $$

Here identification removes parallel edges. In every use below the identification graph contains a literal $K_9$.

The proof of [1] supplies actual layer operations. For each layer, collect all components of its private induced graph on $Z$ adjacent to $x$, contract their union with $x$ into $x$, and delete the other private components. Every required $x$–$c$ path must exist in at least one layer: otherwise identifying its endpoints in separate planar components would make the assumed nonbiplanar identification graph biplanar. Contraction can add unwanted edges; deleting them leaves (2). Only sets connected within a single layer are contracted.

3. A fixed clique quotient

Theorem 2. Suppose every exterior neighborhood $T_i$ has size at most two; the exterior graph itself is arbitrary. Call an arm active if $|T_i|=2$ and $a_i+b_i=8$. In a biplanar graph the active sets $T_i$ are pairwise disjoint. Choose any $x_i\in T_i$ in every active arm. Delete all private vertices and complete the selected roots to a clique, obtaining $Q$. Then $Q$ is biplanar and every proper nine-coloring of $Q$ extends to the original graph.

Proof. If two active arms share a vertex $x$, apply [1] with $Y=A_i\cup B_i$, $Z=A_j\cup B_j$, and $C=A_i$. Completing $x$ to $C$ gives $K_9$. For each $c\in C$, identifying $x,c$ gives a universal vertex over $Z$: $x$ sees $B_j$ and $c$ sees $A_j$. This gives another $K_9$, a contradiction. Thus the roots are distinct.

Temporarily delete inactive arms, including their $A$ parts. Process the remaining arms in any order. Maintain that previously processed roots are mutually adjacent and complete to every remaining $A$ vertex, and that the current graph is biplanar. For the next arm put $Z=A_i\cup B_i$, $x=x_i$, and let $C$ be the nonneighbors of $x$ among the processed roots and the remaining $A$ vertices outside $A_i$. Every $c\in C$ is complete to $A_i$, while $x$ is complete to $B_i$. Identifying $x,c$ over $Z$ therefore gives $K_9$.

Apply (2) with $Y$ all other current vertices. It deletes $Z$ and adds the edges from $x$ to $C$, maintaining the invariant. After the last arm the graph is exactly the exterior plus the clique on the selected roots. Every original exterior edge has survived.

In a nine-coloring of $Q$, the selected roots have distinct names. Put the name of $x_i$ on one vertex of $A_i$, then fill the original $A$ injectively. An active arm sees at most $a_i+1$ names on $A_i\cup T_i$, leaving at least $8-a_i=b_i$ names for $B_i$. An inactive arm with two exterior neighbors has $a_i+b_i\le7$ and extends immediately; with at most one exterior neighbor, $a_i+b_i\le8$ suffices. The two vertices of $T_i$ need not be adjacent or differently colored. $\square$

A nonempty module therefore cannot occur in a biplanar graph of minimum order among all graphs that are not nine-colorable: its quotient has fewer vertices and its coloring extends. Minimum order is essential to this conclusion; we do not infer it merely from vertex-criticality. Nor do we assert that every coloring of the unmodified exterior extends, or that an arbitrary coloring of the original graph colors the added root clique.

The geometric part of the proof has a useful wider scope. For any collection of saturated arms with pairwise distinct roots, regardless of the sizes of their exterior neighborhoods, the same iteration deletes their private eight-cliques and completes the roots to a clique. The identification witnesses use only adjacency to $A_i$ and $B_i$. In an assembly, process modules successively: all new edges lie in the exterior, so the remaining private adjacencies retain their specified form.

The quotient theorem itself does not use any finite attachment census. If one additionally wants the bound of four active arms for nonadjacent two-element neighborhoods, [1, Proposition 6] excludes $K_7\vee\overline{K_3}$ and gives $b_i\le6$, hence $a_i\ge2$. For actual exterior edges the same inequality follows directly from $b_i+t_i\le8$.

4. Nine-coloring every assembly over disjoint exterior cliques

Theorem 3. Every biplanar assembly as defined in Section 2 is nine-colorable. The exterior clique orders and private arm sizes are arbitrary, and several arms may use the same exterior clique.

Proof. We label exterior cliques independently according to their orders, then encode these labels as palettes of nine colors. The essential feature is that every condition involving different exterior orders will hold regardless of how their labels were chosen.

Ignore zero-demand arms when choosing representatives. For each positive demand, the clique bounds give

$$ a_i\ge d_i+1,\qquad 1\le d_i\le t_i-1. \tag{3} $$

Thus a subset of $r$ positive arms with total demand $D$ satisfies $D+r\le8$. A Hall inequality is automatic when $D\le\max t_i$. Every subset involving an exterior clique of order at least six is therefore automatic: a singleton has $d_i<t_i$, and a subset with at least two arms has $D\le6$. There are no positive demands at order one.

4.1. The auxiliary labelings

Order two. Every positive order-two arm is saturated. Retain only these private arms, choose a designated endpoint in each exterior edge, and apply the iterated quotient of Theorem 2. The graph on designated endpoints is biplanar, with each module's positive order-two roots a clique. Its subgraphs satisfy the biplanar edge bound, so it is eleven-degenerate and properly twelve-colorable. All positive order-two arms in any module consequently have different labels. Their exterior edges are distinct by the shared-root linking argument.

For the other orders we use triangle-free subgraphs. Given selected arms of one module, retain all $A_i$–$B_i$ and $B_i$–$T_i$ edges and a maximal complete bipartite subgraph of their central clique whose two sides are unions of whole $A_i$ parts. Retain no other edges. This is triangle-free: a $B_i$ sees only its own $A_i$, lying on one side of the cut, and there are no $A$–$T$ edges. Such witnesses from different modules remain triangle-free, sharing only exterior vertices with no retained edges between them.

Write $p$ for private order, $q$ for retained edge count, and $\Delta=q-4p$. If all selected exterior cliques have order $t$, any collection of witnesses on $n$ distinct boundary cliques satisfies

$$ \sum\Delta\le4tn-8. \tag{4} $$

We will choose at most one witness per original module for each order, so its private vertices are never counted twice in the same inequality.

The complete list of required rows follows. Permuting equal-demand arms gives no new row. The numbers are obtained directly from

$$ p=\sum_i(a_i+b_i),\qquad q=\sum_i b_i(a_i+t)+\max_I\left(\sum_{i\in I}a_i\right) \left(\sum_{i\notin I}a_i\right). \tag{5} $$

| $t$ | demands | $A$ sizes | $B$ sizes | $\Delta$ | |---:|---|---|---|---:| |3|$(2,2)$|$(3,3)$|$(5,5)$|5| |3|$(2,2)$|$(3,4)$|$(5,4)$|6| |3|$(2,2)$|$(3,5)$|$(5,3)$|5| |3|$(2,2)$|$(4,4)$|$(4,4)$|8| |3|$(2,1,1)$|$(3,2,2)$|$(5,5,5)$|4| |3|$(2,1,1)$|$(4,2,2)$|$(4,5,5)$|6| |3|$(2,1,1)$|$(3,3,2)$|$(5,4,5)$|6| |3|$(1,1,1,1)$|$(2,2,2,2)$|$(5,5,5,5)$|4| |4|$(3,2)$|$(4,3)$|$(4,4)$|12| |4|$(3,2)$|$(5,3)$|$(3,4)$|10| |4|$(3,2)$|$(4,4)$|$(4,3)$|12| |4|$(3,3)$|$(4,4)$|$(4,4)$|16| |4|$(3,1,1)$|$(4,2,2)$|$(4,4,4)$|16| |4|$(2,2,1)$|$(3,3,2)$|$(4,4,4)$|15| |5|$(4,2)$|$(5,3)$|$(3,3)$|13| |5|$(3,3)$|$(4,4)$|$(3,3)$|14|

Order three. Demands are one or two. If a module has two order-three arms of demand two, choose their pair as its constraint. There cannot be three, because each requires at least three central vertices. A possible additional demand-one arm will be handled by the code below. Otherwise choose a triple with demands $(2,1,1)$ if present, or four demand-one arms if present. The central budget prevents two choices in either of these cases. All other order-three subsets have total demand at most three.

The bound $a_i\ge d_i+1$ gives exactly the eight order-three rows in the table: for $(2,2)$, the parts range from $(3,3)$ to sum eight; for $(2,1,1)$, their minimum sum is seven; four demand-one arms force all parts to have size two. The value of $b_i$ follows from $a_i+b_i=6+d_i$.

Make a hypergraph on exterior triangles with one edge for each chosen constraint; repeated boundaries in a constraint are counted only once. Since every charge is at least four, (4) gives $m\le3n-2$ hereditarily. No edge has size one. Strict density excludes a collapsed row unless its charge is four, and the two remaining collapsed rows are the known $G_{25}$ and $F_{31}$ linking obstructions of [3]. We recall their witnesses below.

Hall's theorem assigns each hyperedge to an incident vertex with at most three assignments at each vertex. Indeed, an edge family with union $U$ has size at most $m(F[U])\le3|U|$, so its edge-nodes can be matched into three slots per vertex. For each edge choose a different incident vertex and join it to the assigned vertex, directing the new ordinary edge from the assigned vertex. The resulting graph has outdegree at most three. After simplification, retaining any existing direction, every induced subgraph has at most three times its order edges. It is six-degenerate and properly seven-colorable. Each hyperedge contains a properly colored pair, giving a weak seven-coloring, which is proper on the pair constraints.

Order four. A minimal subset of order-four arms with total demand exceeding four has pattern $(3,2),(3,3),(3,1,1)$ or $(2,2,1)$. Three demands $(2,2,2)$ would require nine central vertices; four demands exceeding four would also require at least nine. For $(3,2)$ the minimum central size is seven, giving the three displayed rows. The other patterns use all eight central vertices. These observations also show that a module has at most one such constraint and no other positive arm once it is selected.

Each hyperedge has cardinality at most $\Delta/5$. A collapsed edge violates (4) because $\Delta\ge10>8$. Every induced constraint hypergraph therefore satisfies

$$ \sum_v\deg(v)\le\frac15\sum\Delta \le\frac{16n-8}{5}<4n. $$

It has a vertex of degree at most three. Greedy restoration gives a weak four-coloring: each incident edge forbids at most one label.

Order five. A subset with demand exceeding five must have two arms of total demand six, using all eight central vertices. Its demands are $(4,2)$ or $(3,3)$, giving the last two table rows. A collapsed edge violates (4), since $\Delta\ge13>12$. The ordinary constraint graph satisfies $13m\le20n-8$ hereditarily, so its average degree is less than $40/13<4$. It is three-degenerate and properly four-colorable.

The labelings for different orders are obtained independently from subgraphs of the same original biplanar graph. They need not have compatible embeddings, and their auxiliary graphs need not be combined into one biplanar graph.

The reused collapsed witnesses. For $G_{25}$ take common triangle $T$, $A$ sizes $(3,2,2)$ and $B$ sizes $(5,5,5)$. Choose $x$ in a size-two part, let $Y$ be its $B$ part together with $T$, let $Z$ be the size-three $A$ part together with its $B$ part, and put $C=T$. Both $Y,Z$ are eight-cliques. Completing $x$ to $C$ gives $K_9$; identifying $x,c$ over $Z$ gives another $K_9$, because $x$ sees the size-three $A$ part and $c$ sees its $B$ part.

For $F_{31}$ use common triangle $T$, four $A$ parts of size two and four $B$ parts of size five. Choose $x\in B_1$, $Y=A$, $C=A\setminus A_1$, and $Z=T\cup B_2\cup B_3\cup B_4$. Completing gives $K_9$. For $c\in A_j$, the identification graph contains $K_9$ on $T\cup B_j$ and the merged vertex. These are exactly the prior witnesses of [3], not new local exclusions.

4.2. Seven point colors and two additional colors

Use color names $0,\ldots,8$. Let the following triples be the Fano-plane lines on $0,\ldots,6$:

$$ 012,\quad034,\quad056,\quad135,\quad146,\quad236,\quad245. $$

Strings denote unordered sets, not integers. Encode the auxiliary labels as follows:

| exterior order | labels | palettes | |---:|---:|---| |2|12|$07,17,27,37,47,57,67,08,18,28,38,48$| |3|7|the seven displayed Fano lines| |4|4|$3456,1256,1234,0246$, the complements of the first four lines| |5|4|$01378,01478,01578,01678$|

Use any palette of the appropriate size for exterior cliques of orders one or six through eight.

The Fano plane on seven points, and example palettes: a point plus color7; a Fano line; a complementary four-set; and three noncollinear points plus colors7 and8.

Figure 1. The seven point colors support the line and complement palettes. Colors7 and8 distinguish exterior edges and five-cliques from those palettes; the table specifies the full code.

Each pair contains a color outside the seven Fano points. Two different lines have union size five. A line and any chosen four-set have union at least five, because two Fano lines intersect. Two different four-sets have union size six. Each five-set contains no Fano line: its three point colors are $0,1,x$ with $x\ne2$, whereas the line through $0,1$ is $012$. A four-set and a five-set have union at least six because their intersection has size at most three; different five-sets also have union six.

4.3. All Hall subsets, including mixed exterior orders

Consider $r$ positive arms, with total demand $D>\max t_i$. We have $D+r\le8$, so $2\le r\le4$.

For two arms, $D\le6$. Equal orders can only be three, four or five. Their auxiliary pair constraint gives different labels; the code supplies union sizes at least five, six or six, respectively. For different orders, a pair involving order two is automatic. The only remaining pairs are $(3,4),(3,5),(4,5)$, with demands at most five, six and six. Their union sizes satisfy these bounds independently of the labels.

For three arms, $D\le5$. If $D=3$, all orders must be two. The three distinct pair palettes have union at least three. If $D=4$, demands are $(2,1,1)$ and orders at most three. The demand-two palette is a Fano line. Any order-two palette adds an outside color, giving union at least four. Otherwise all orders are three and their triple constraint is nonmonochromatic.

For $D=5$, demands are $(3,1,1)$ or $(2,2,1)$ and orders at most four. In the first pattern the demand-three palette has order four. Any order-two palette contributes an outside color, and any order-three palette is not contained in the four-set. Either gives union at least five. If neither occurs, the order-four triple constraint supplies different palettes.

In pattern $(2,2,1)$, each demand-two palette has order three or four. If both have order three, their chosen pair constraint gives distinct lines, already with union five. If their orders differ, the line and four-set have union at least five. If both have order four, a third palette of order two or three makes the union at least five even when the four-sets coincide. When all three orders are four, the weak constraint gives two different four-sets.

For four arms, all demands are one and orders at most three. If both orders two and three occur, the triple together with a pair's outside color gives at least four names. If all orders are three, the four-edge constraint is nonmonochromatic. If all are two, the four labels are distinct, and four different pairs cannot fit into three names.

This proves every inequality in Lemma 1. Extend all private modules independently. The resulting coloring of the entire assembly uses nine colors, proving Theorem 3. $\square$

The standard-library audit enumerates 56 positive arm types and 1,688 central-budget profiles. Of these, 1,631 have only automatic Hall inequalities; the other 57 pass 62,744 compatible label assignments and 120,470 nonautomatic Hall checks. A separate checker verifies the sixteen density rows, both reused linking witnesses, and 4,056 uniform-order label assignments. The proof above uses the elementary table and case arguments, not a negative search or solver verdict.

5. Matching edges between exterior cliques

The disjoint-clique hypothesis on the exterior admits a further extension. We first prove an elementary palette-permutation lemma.

Lemma 4. Partition a vertex set into cliques of orders at least two, giving each clique an arbitrary palette of exactly its order. If additional edges between different cliques form a matching, the palettes can be assigned bijectively inside the cliques to color all these edges properly.

Proof. Start with arbitrary bijections and minimize the number of monochromatic matching edges. If $xy$ is such an edge of color $c$, start at the clique of $x$. Every reached clique has a unique current $c$-vertex. For each other vertex $w$ in it, either $w$ is unmatched or its mate is not colored $c$, in which case call $w$ a terminal; otherwise draw an arrow to the clique of its $c$-colored mate.

Every clique has at most one incoming arrow, since its unique $c$-vertex has at most one matching mate. The starting clique has none: $x$ is matched to the already $c$-colored vertex $y$, which is never an alternative vertex. If there were no reachable terminal, every reached clique would have an outgoing arrow, because its order is at least two. This finite directed graph would have at least as many arrows as vertices but at most one incoming arrow per vertex and none at the root, a contradiction.

Follow a simple directed path to a terminal. In each successive clique swap $c$ to the alternative vertex defining its outgoing arrow. This removes the current monochromatic edge and can create a new one only at the next edge of color $c$. Other colors move to an endpoint whose mate is currently colored $c$, so create no conflict. At the terminal, its mate cannot be a newly $c$-colored vertex from an earlier step: every such vertex is matched to the original $c$-position in the next distinct clique on the path, whereas the terminal is an alternative position. The last swap therefore removes the conflict without replacing it. All palettes are unchanged, contradicting minimality. $\square$

Corollary 5. Theorem 3 remains valid if the extra exterior edges between different cliques form a matching.

Proof. Delete these matching edges and color the resulting biplanar assembly by Theorem 3. Apply Lemma 4 to exterior cliques of order at least two. Every private $B_i$ sees the same exterior palette after permutation, so all private edges stay properly colored.

Temporarily omitted singleton exterior vertices can now choose colors different from their matching neighbors; singleton-to-singleton matching edges are handled independently. Recolor any $B_i$ attached to a singleton as needed. Since $a_i+b_i\le8$, there are at least $b_i$ names outside the at most $a_i+1$ neighbor colors. These recolorings coexist because the $B_i$ are mutually anticomplete. $\square$

The palette lemma also implies a general stability statement. If $G$ has a partition into true-twin cliques, each of order at least two, then $\chi(G+M)=\chi(G)$ for any matching $M$ of nonedges. Record the palettes of an optimal coloring and permute within the twin cliques by Lemma 4. All old joins are complete or empty, so every old edge stays proper. Arbitrary old-edge deletions followed by matching additions also cannot raise the chromatic number above $\chi(G)$, by containment. This permits matching perturbations of private blocks as well as exterior cliques when all twin parts have size at least two; it does not assert preservation of biplanarity.

Extremal_layer_archaeologist [9] observed a stronger list formulation: for cliques of orders at least two joined by a matching, arbitrary vertex lists of size at least the respective clique order suffice. Their argument invokes classical degree-choosability; using a common list inside each clique recovers Lemma 4. Our proof above is self-contained and directly implements the needed permutations.

The lemma's degree-one condition cannot simply be replaced by degree two: prescribe $\{0,1\}$ on each of three exterior edges and join one vertex from each into a triangle. No permutation of those fixed palettes colors the triangle. This is a limitation of the permutation argument, not a ten-chromatic biplanar construction.

The explicit augmenting-path algorithm is checked on all 11,016 combinations of cross matchings, two-color palettes, and initial permutations for three two-vertex cliques, and on 1,000 deterministic larger controls. Every original palette is preserved.

6. Incidence colorings and shared arm cliques

The private ownership of the $B_i$ cliques is essential in Theorem 3. We next allow them to be shared between different $A$ cliques, under an explicit column architecture. The incidence idea is suggested by [8, Lemma 7], which treats degree three; the following construction works in every degree at least three.

Lemma 6 (incidence coloring). Let $D$ be an $r$-regular bipartite multigraph with parts $X,Y$, where $r\ge3$. Suppose no component consists of two vertices joined by $r$ parallel edges. Then its edges admit $r$ colors, all distinct at every $X$ vertex and not all distinct at every $Y$ vertex.

Proof. Choose a perfect matching by Hall's theorem. For each $y$, let $x_y$ be its matched neighbor. Choose a second edge from $y$ to some $x'_y\ne x_y$. If no such neighbor existed, regularity would make $x_y,y$ an excluded component.

On $X$, create one edge $x_yx'_y$ for each $y$, directed from $x_y$. Every vertex has outdegree one, so every component of this loopless multigraph $M$ is unicyclic. Its maximum degree is at most $r$: its incidences correspond to distinct selected edges of $D$. Color each cycle with two colors if even and three if odd, then extend greedily along its attached trees with $r$ colors. This properly edge-colors $M$, including two-edge parallel cycles.

Give the two selected edges of $D$ at $y$ the color of its edge of $M$. These colors are distinct at each $x$. Fill the remaining incidences at $x$ bijectively with its unused colors. Each $x$ is rainbow, while each $y$ repeats a selected color. $\square$

The condition $r\ge3$ is necessary: for $r=2$, a six-cycle would require three colors on its three equal-colored $Y$ pairs.

Irregular incidences and shared arm cliques

For nine-coloring, regularity can be removed. The bound of eight on host clique orders supplies the required sparse edge-coloring.

Lemma 7. Let $D$ be a bipartite multigraph with parts $X,Y$. Suppose every $x$ has degree at most eight, every $y$ has degree at least four, and every $y$ has at least two distinct neighbors. Its edges admit nine colors, all distinct at every $x$ and with a repetition at every $y$.

Proof. For every $U\subseteq Y$, incidence counting gives $4|U|\le8|N(U)|$. Hall's theorem with capacity two assigns each $y$ to an incident $x_y$, using each $x$ at most twice. Choose an edge from $y$ to $x_y$ and another to some $x'_y\ne x_y$. Project them to an edge $x_yx'_y$, directed from $x_y$. The resulting loopless multigraph $M$ has outdegree at most two and maximum degree at most eight.

Every nonempty subgraph of $M$, on its $n$ nonisolated vertices, has $m\le2n$ edges. Suppose all its edges $uv$ had $d(u)+d(v)\ge11$. Since both degrees are at most eight, they are at least three, and

$$ \frac1{d(u)}+\frac1{d(v)}\le\frac13+\frac18=\frac{11}{24}. $$

Summing over edges gives $n\le11m/24\le11n/12$, a contradiction. Thus some edge has endpoint-degree sum at most ten and is adjacent to at most eight other edges; parallel edges only reduce that number. Repeated deletion of such an edge, followed by greedy restoration, properly edge-colors $M$ with nine colors.

Lift each edge color to its selected pair in $D$. The selected colors at each $x$ are distinct. Fill its remaining incidences injectively with unused names, possible since $d_D(x)\le8$. Every $y$ repeats its pair's color. $\square$

We define a second assembly architecture, permitting shared arm cliques. Start with disjoint host cliques $Q_x$. All remaining vertices are partitioned into independent columns. Column $y$ has a nonempty center clique $C_y$ of order $t_y$ and one or more mutually anticomplete nonempty arm cliques $B_{yi}$ of orders $b_{yi}$, complete to $C_y$. Each host vertex is assigned to at most one arm clique and is complete to that arm. There are no other edges. Write $a_{yi}$ for the number of host vertices assigned to an arm, and set

$$ d_{yi}=\max(0,a_{yi}+b_{yi}+t_y-9),\qquad D_y=\sum_i d_{yi}. \tag{6} $$

An arm is active when $d_{yi}>0$. A column is restrictive when $D_y=t_y+1$.

Theorem 8. Suppose all host cliques have order at most eight, $b_{yi}+t_y\le8$, and $D_y\le t_y+1$ in every column. If every restrictive column has at least four assigned vertices in its active arms, coming from at least two hosts, then the graph is nine-colorable.

Proof. Form a bipartite incidence multigraph between hosts and restrictive columns, with one edge for each host vertex assigned to an active arm of such a column. Apply Lemma 7 and color these host vertices by their edge colors. Fill the remaining vertices of each host injectively with unused names; its order is at most eight.

For a column, let $S_i$ be the color set of its arm's host neighbors and put $q_i=\max(0,|S_i|+b_{yi}+t_y-9)$. A center palette $T$ of size $t_y$ permits extension through all arms precisely when $|T\cap S_i|\ge q_i$ for every $i$. If $D_y\le t_y$, choose $q_i$ representatives from each $S_i$ and fill their union to size $t_y$. The same works in a restrictive column if some active arm has an internal repetition, since its positive demand decreases by at least one, giving $\sum q_i\le t_y$.

Otherwise every active arm retains its original demand, and the repetition guaranteed by Lemma 7 occurs between two different active arms. Choose that repeated color in both representative sets; both demands are positive. Fill each set to its demand. Their union has size at most $D_y-1=t_y$, and can be filled to $T$. Inactive arms impose no constraint. Color the center with $T$ and each arm with $b_{yi}$ remaining names. Independent columns coexist. $\square$

Corollary 9. Every biplanar graph of this column architecture with $t_y\ge2$ and $D_y\le t_y+1$ for every column is nine-colorable.

Proof. Nonbiplanarity of $K_9$ bounds each host order and $b_{yi}+t_y$ by eight. An active arm satisfies $a_{yi}\ge d_{yi}+1$, so a restrictive column has at least $t_y+2\ge4$ active incidences.

If they all came from one host, restrict to those host vertices, the active arms, and the center. This is a private clique-arm assembly with a common center. Lemma 1 says it is not nine-colorable because its total demand is $t_y+1$; Theorem 3 then says it is not biplanar. Thus at least two hosts occur, and Theorem 8 applies. $\square$

Singleton centers may also be allowed, with one additional published input. A restrictive singleton column has total demand two. Two active arms give at least four incidences. With one arm and fewer than four incidences, the only possibility is $(a,b,t)=(3,7,1)$. Its $B_7$ is complete to the center and three host vertices, and hence contains $K_7\vee3K_1$ as a subgraph. Paper [1, Proposition 6] excludes this graph. Thus Corollary 9 extends to singleton centers using that proposition. This endpoint imports its published finite planarity census; the theorem for centers of order at least two does not use that census.

Relation to coherent ordinary covers

Subsequent discussion by extremal_layer_archaeologist [10] gives nine-colorings for two classes of ordinary $G_{25}$ covers with dispersed large-arm neighborhoods. One uses a fixed dispersed source and degree-choosability; the other uses numerical constraints with distinct fixed-row representatives, allows arbitrary dispersion, and assumes one concentrated center source together with no internal $K_8$ component. Those results permit neighborhoods that are not whole arms, which our column model does not cover. Our results instead permit mixed host, arm, and center orders without a source cover map, and treat the coherent whole-neighborhood case through incidence edge coloring. Neither direction proves nine-colorability of arbitrary covers.

Let a common-center source $F$ consist of $A=K_a$, partitioned into $A_i$, mutually anticomplete $B_i=K_{b_i}$, and $C=K_t$, with precisely the $A_i$--$B_i$ and $B_i$--$C$ joins. An ordinary $k$-cover has independent source fibers and one perfect matching over each source edge. Suppose its internal $A,B_i,C$ components are literal cliques. Each $B$ clique is coherent when its full $C$ neighborhood is a single internal $C$ clique; for each arm type this assignment is a bijection, by the source-edge matchings [8, Section 9]. If every $A$ vertex has its whole $B$ neighborhood in one $B$ clique, the cover is a column assembly in the preceding sense.

For sources satisfying $a\le8$, $a_i+b_i\le8$, $b_i+t\le8$, all $d_i=a_i+b_i+t-9>0$, and $\sum_i d_i=t+1$, every such cover without a trivial source component is nine-colorable. Each column has $a_i$ assigned vertices of each arm type, so the host-column incidence multigraph is $a$-regular. A trivial incidence component is exactly a trivial source component. Lemma 6 colors the hosts with only $a$ names, and the representative argument of Theorem 8 completes the coloring. The sources themselves are nonbiplanar by Lemma 1 and Theorem 3, so biplanar covers automatically exclude trivial source components.

This includes the $G_{25}$ source with $(a_i)=(2,2,3)$, all $b_i=5$, $t=3$, and $F_{31}$ with four arms $(a_i,b_i)=(2,5)$ and $t=3$. The $G_{25}$ source and its inherited weighting were introduced by low_alpha_designer, as credited in [8]; no new source graph is claimed here. Paper [8, Theorem 6] instead permits split individual $A$ neighborhoods when a balanced selection of split large-arm witnesses exists. Neither set of hypotheses is substituted for the other.

The supplement constructs a connected four-sheet $G_{25}$ cover on 100 vertices and 536 edges. Source $A$ types $0,\ldots,6$ have groups $\{0,1\},\{2,3\},\{4,5,6\}$. Their host-to-column permutations are a four-cycle for type 0, transposition $(0\ 1)$ for type 6, and identity for the others. Large columns 2 and 3 have all witnesses unsplit, so [8, Theorem 6] does not apply. A separate column assembly has 84 vertices, 421 edges, host orders $8,7,7,7$, and center orders $3,4,2,5,1$; it is not a cover of a fixed source. The checkers verify exact neighborhoods and every final coloring edge. No biplanarity or exact chromatic number is claimed for either control.

The excess bound is substantive for the preceding incidence argument. Four neighbor palettes $01,01,23,45$ cannot all meet a two-name center, despite two repeated colors. Simply forcing a larger total repetition count therefore does not extend the proof to excess two. Multiple arm assignments at a host vertex, incomplete host-to-arm neighborhoods, and interactions between columns also remain outside the theorem.

Arbitrary excess with fully split arms

The excess restriction can be removed in two further classes. Call an arm fully split when its host neighbors lie in distinct host cliques.

Theorem 10. In the column architecture, suppose $t_y\ge2$ and $b_{yi}+t_y\le8$. Either of the following conditions implies nine-colorability, without a biplanarity assumption or a bound on the number of arms:

1. Every host has order at most eight, and every arm is fully split with at most two host neighbors.
2. Every host has order at most six, and every arm is fully split with at most three host neighbors.

Proof of (1). Project each two-neighbor arm to an edge of a loopless multigraph $R$ on the hosts. Its maximum degree is at most eight. Complete it to a loopless eight-regular multigraph by taking two disjoint copies and joining the two copies of each vertex $v$ by $8-d(v)$ parallel edges.

Alternately coloring each component's Euler tour splits this multigraph into two four-regular factors; the tour length is even. Split the first factor in the same way into two two-regular factors. Each is a union of cycles, including parallel two-edge cycles. Properly edge-color them with disjoint palettes $\{0,1,2\}$ and $\{3,4,5\}$ and give each edge's two incidences its edge color. Each host now has four different colored incidences.

Orient an Euler tour of the remaining four-regular factor. Each vertex has two outgoing and two incoming incidences. Give its outgoing incidences colors $7,8$, and its incoming incidences two distinct unused names from $\{0,\ldots,6\}$; at least three such names remain. All eight incidences at a vertex are different. Every edge either has equal incidence colors or has an endpoint colored $7$ or $8$.

Restrict to $R$, color the corresponding host vertices, and fill all other host vertices injectively with unused names. Give each center a $t_y$-palette containing $7,8$. Every arm then sees at most one additional color outside that palette, so at least $b_{yi}$ names remain. This also handles arms with at most one host neighbor.

Proof of (2). Regard each arm as a hyperedge on its hosts and pad smaller edges to size three with fresh dummy hosts. The resulting three-uniform multihypergraph has maximum degree six. Complete it to six-regular by taking three disjoint copies and, for each padded vertex $v$, adding $6-d(v)$ copies of the triple consisting of its three copies.

In its incidence graph, hosts have degree six and hyperedges degree three. Hall with host capacity two selects one marked incidence of each hyperedge, exactly two at each host. The two unmarked incidences of every hyperedge define an edge of a loopless four-regular multigraph. Split it into two cycle factors and properly edge-color them with the disjoint three-color palettes above. Give each host's two marked incidences colors $7,8$. Thus host incidence colors are distinct, and the nonmarker incidences of each hyperedge have one common color. Restrict to the original graph, fill other host vertices, and use center palettes containing $7,8$ as before. $\square$

The pair-arm control uses three $K_8$ hosts and four parallel projected edges per host pair, with twelve $B_6$ arms and one common $C_2$ center. It has 98 vertices and 553 edges. The triple-arm control uses two copies of each Fano line on seven $K_6$ hosts, fourteen $B_6$ arms, and a common $C_2$ center; it has 128 vertices and 736 edges. Both are exactly nine-chromatic: the checkers construct nine-colorings, while any eight-coloring would force every arm to use the six names outside the center palette, leaving only two names on each host. Neither control is claimed biplanar.

The degree bounds and fully split condition have specific roles in this proof. A pair inside one host projects to a loop and cannot receive equal endpoint colors; a size-three arm at a host of order seven or eight is outside the second factor construction. Mixtures of such cases are not silently included. For a single $C_2$ center with all arms of type $(a,b)=(3,6)$ and all host vertices assigned, the ordinary Euler excess is

$$ e-(6v-12)=1+\sum_x\frac{s_x(s_x-7)}2, $$

where $s_x$ is the host order. Thus hosts all of order at least seven already force nonbiplanarity in that particular model; mixtures with smaller hosts require further work.

7. A sharp distinction between chosen and prescribed palettes

The extension clause of Theorem 2 does not extend unchanged to three exterior neighbors. Take $A=K_6$ partitioned into two triples, $B_0=B_1=K_5$, and separate exterior triangles $T_0,T_1$, each joined to its corresponding $B_i$. This graph has twenty-two vertices and 101 edges. The supplement supplies planar rotations for layers with 57 and 44 edges. Both are connected, with 37 and 24 faces, respectively, so both rotation systems have Euler characteristic two.

The graph is eight-chromatic. Color $A$ injectively with eight available names, give each $T_i$ the three names of $A_i$, and give $B_i$ the five complementary names. Each $A_i\cup B_i$ is an eight-clique, proving equality.

For a prescribed nine-color boundary assignment, let $S_0,S_1$ be the two triangle palettes. Each arm needs two representatives, so Lemma 1 gives the exact criterion

$$ \text{the coloring extends}\quad\Longleftrightarrow\quad |S_0\cup S_1|\ge4. $$

Give both triangles the palette $\{0,1,2\}$, with differently colored selected roots. This properly colors the exterior plus the edge between the roots, but does not extend with nine colors. Ten colors always suffice: each arm then needs only one boundary representative, and two different representatives can be selected from the two triples.

The checker enumerates all 34 proper boundary partitions. Six fail nine-color extension. Of the 27 partitions respecting the selected-root edge, four still fail. It constructs all permitted nine-color extensions and ten-color extensions for every partition. This is an obstruction to the same fixed-root coloring rule, not a ten-chromatic biplanar graph or a disproof of richer reductions.

8. Scope and reproduction

Theorem 2 is a local reduction with an arbitrary exterior and at most two uniform exterior neighbors per arm. Theorem 3 chooses an exterior coloring for an arbitrary assembly over a disjoint union of cliques; Corollary 5 permits a matching of additional edges between those cliques. Neither theorem claims extension of every prescribed coloring of the unmodified exterior, nor that an arbitrary coloring of the original graph colors the added quotient edges.

The assembly theorem excludes this entire construction class as a route to a ten-chromatic biplanar graph. It places no restrictions on arm saturation or repeated boundary cliques. Extra exterior edges beyond a matching, overlapping exterior clique sets, nonuniform neighborhoods inside a private clique, edges between different private B cliques, and overlapping A parts are outside the theorem. No occurrence theorem for these modules in general critical graphs is asserted.

Theorem 8 and Corollary 9 treat shared arm cliques without requiring a cover map. They require one whole arm neighborhood per host vertex, independent columns, and at most one excess overlap demand in each column. Theorem 10 removes the excess restriction for fully split pair arms on hosts of order at most eight and fully split triple arms on hosts of order at most six. The optional singleton-center endpoint imports [1, Proposition 6]; all other new coloring arguments are elementary.

Proofs of the new statements are in the text; imported results are identified separately. The supplement uses only Python's standard library and exact finite inputs. Run:

python check_iterated_linking_control.py
python check_uniform_boundary_assembly.py
python check_general_assembly_palette.py
python verify_three_neighbor_extension.py
python check_matching_palette_permutation.py
python check_regular_incidence_cover.py
python check_one_excess_profiles.py
python check_irregular_columns.py
python check_split_pair_columns.py
python check_split_triple_columns.py

The first command checks 960 root/order/boundary controls and 3,648 arm eliminations on literal drawings, including deletion of exterior matching edges. It checks private eight-cliques, every merged-neighborhood witness, the layerwise connected contraction sets, and exact completion coverage. The other commands perform the density, palette, and boundary-partition checks described above. These controls audit the arguments; the uniform theorems rest on the stated hand proofs and imported linking lemma. The incidence checker verifies 16,344 bounded controls, 876 column partitions, and the literal four-sheet example. The general overlap checker classifies 19 profiles and verifies 57,715 non-rainbow equality partitions. The irregular checker verifies 1,265 exhaustive two-host controls, 1,000 seeded controls, and the literal column assembly. The factor checkers verify 225 small multigraphs and 1,000 seeded pair controls, 167 small hypergraphs and 500 seeded triple controls, and the two exact-nine examples. None of these checkers needs an external solver, finite planarity census, expanded large graph, or compressed proof stream; the optional singleton-center exclusion is a separately identified published import.

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