Nine-coloring extensions over five-cycle cores and their nineteen-vertex obstructions

1. Problem and contribution

The Earth–Moon problem asks whether a graph whose edges partition into two planar graphs can require ten colors. The order-nineteen result [1] leaves twenty as the first possible order. At that order, biplanarity implies at most $108$ edges and excludes $K_9$ [2], while ten-criticality implies minimum degree nine. We study coloring extensions under these weaker edge and clique restrictions.

For positive integers $a_0,\ldots,a_4$, let $H(a)$ have five clique parts $V_i$ of sizes $a_i$, with complete joins between $V_i$ and $V_{i+2}$ and no joins between adjacent parts; indices are modulo five. Its complement is an independent-set blowup of a five-cycle. The choice of distance-two joins is a labeling convention.

Throughout the extension theorems, $G$ has twenty vertices, is $K_9$-free, has at most $108$ edges, and has an independent triple $S$ whose vertices each have degree at least nine. A spanning copy of $H(a)$ in $G-S$ permits arbitrary additional core edges. No minimum degree is imposed on the core, and neither biplanarity nor criticality is assumed unless explicitly stated.

The three source graphs

$$J_1=H(3,3,3,5,5),\qquad J_2=H(3,3,4,5,4),\qquad J_3=H(3,4,4,4,4)$$

are known from Trivedi [3, Table 1 and Section 5]. We use their identities and elementary nonbiplanarity as prior work. Our new results concern twenty-vertex extensions, which need not themselves be five-cycle blowups.

Theorem 1 (coloring extensions). Under the hypotheses above, suppose $G-S$ contains the spanning core in a row of the following table. Then $G$ is nine-colorable if and only if it contains none of the graphs in the second column as a subgraph. An empty second column means every such extension is nine-colorable.

| Core $H(a)$ | Forbidden nineteen-vertex family | Core edges | |---|---|---:| | $(1,2,2,6,6)$, equivalently $F(2,6)$ | empty | 76 | | $(2,4,3,4,4)$ | empty | 80 | | $(2,3,3,4,5)$ | empty | 79 | | $(3,3,3,3,5)$ | empty | 79 | | $(2,3,2,5,5)$ | empty | 79 | | $(3,3,4,3,4)$ | $J_3$ | 79 | | $(2,3,3,5,4)$ | $J_1$ | 78 | | $(2,3,4,4,4)$ | $J_3$ | 78 | | $(3,3,3,4,4)$ | $J_1,J_2,J_3$ | 78 | | $(2,3,4,5,3)$ | $J_2$ | 77 | | $(2,2,2,5,6)$ | empty | 76 |

For the two rows whose family is only $J_3$, every noncolorable extension actually contains an induced $J_3$. The other containment statements require only subgraphs. In particular every biplanar extension in the table is nine-colorable. A noncolorable extension in any row cannot be ten-vertex-critical, since it contains a proper subgraph already requiring ten colors.

Theorem 2 (structural transfer). Under the same hypotheses, let $G-S$ contain any spanning $H(a)$ with positive part sizes summing to seventeen. Then every nineteen-vertex induced subgraph of $G$ has an independent triple if and only if $G$ contains none of $J_1,J_2,J_3$ as a subgraph. There are precisely twenty-six possible part types up to dihedral symmetry under the edge and clique bounds. In all thirteen types with a singleton part, the nineteen-vertex condition holds automatically.

Theorem 2 does not assert nine-colorability for every one of those twenty-six core types. Theorem 1 closes ten of the thirteen nonsingleton types and one singleton type. Other critical cores on seventeen vertices and other core orders remain outside these statements.

2. Prior work and distinction

The weighted extension approach and the order-nineteen reduction were developed in [1]. Conditional coloring certificates already gave a full spanning-$F(3,4)$ exclusion and a degree-fourteen slice for $F(2,2)$ [4]. The eleven-vertex critical-core branch was treated separately in [5]; absorbing a further outside vertex yields sixteen-core results in [6]. Here $F(a,b)$ consists of two disjoint eight-cliques, exceptional subsets of sizes $a,b$ joined completely across them, and a final vertex adjacent to their complementary subsets. It has $71+(a-1)(b-1)$ edges. The first row of Theorem 1 is a new full extension of this exceptional-family analysis.

The nonsingleton cores have no two disjoint eight-cliques, so they lie outside that exceptional branch. The methods used to obtain numerical certificates are extensions of [1,4], not a new optimization principle. The advances are the complete additional core exclusions, exact relaxed obstruction classifications, and the minimal-edge reduction underlying Theorem 2.

Trivedi [3] proves nine-colorability for biplanar graphs that are themselves clique blowups of a five-cycle. Its Table 1 and Section 5 already identify our three nineteen-vertex sources; its adjacent-join vectors are obtained from ours by $w_j=a_{2j\bmod5}$. Trivedi's Theorem 3.5 supplies the elementary cross-edge obstruction used below. Our arbitrary twenty-vertex augmentations need not be blowups and are not covered by that result.

Szeider [7] treats biplanar graphs of independence number two more generally. That stronger theorem would also exclude the known sources, but is unnecessary for the explicit biplanar corollaries here. The smaller cross-edge argument makes their dependence clear.

3. Core and source structure

Every $H(a)$ in the table has independence number two. Hence a seventeen-vertex core needs at least nine colors. After deleting a vertex, write $b_i$ for the remaining part sizes, summing to sixteen, and set

$$x_i=8-b_{i+2}-b_{i+4}.$$

The original clique bounds $a_i+a_{i+2}\leq8$ make these integers nonnegative. Direct substitution gives $x_{i-1}+x_i=b_i$. Allocate $x_i$ vertices from each pair of adjacent parts and match them across those parts in the complement. This produces an eight-coloring after every vertex deletion. Thus the cores are nine-vertex-critical. This fact describes their origin; the extension proofs allow noncritical supergraphs obtained by adding core edges.

Every clique of $H(a)$ occupies at most two nonadjacent parts. If all five parts have size at least two, two disjoint cliques cover at most fifteen vertices, so no two disjoint eight-cliques exist.

For completeness, the known source classification [3] has a short deficit description in our indexing. Set $y_i=8-a_i-a_{i+2}$. At order nineteen, $y_i\geq0$, $\sum_i y_i=2$, and

$$a_i=3+y_{i+1}+y_{i+2}.$$

There are three dihedral types of nonnegative five-vectors of sum two: one entry two, two adjacent entries one, or two nonadjacent entries one. These yield precisely $J_1,J_2,J_3$, with $98,98,99$ edges. This is a rederivation of prior work, not a classification of all nineteen-vertex critical graphs.

Each source has independence number two and chromatic number ten. Deleting a vertex gives nine colors by the same allocation formula with $9$ in place of $8$. The supplied source checker also records a proper nine-coloring after every edge deletion. For an edge $uv$, it finds a common complement-neighbor $w$ and a perfect matching of the remaining sixteen vertices in the complement; after deleting $uv$, the triple $\{u,v,w\}$ receives one color and the eight pairs receive the others.

Deleting within-part edges of $J_1,J_2,J_3$ leaves triangle-free graphs with $69,70,72$ edges. A triangle-free biplanar graph on nineteen vertices has at most $2(2\cdot19-4)=68$ edges. Trivedi's criterion [3] therefore excludes all three sources from a biplanar graph.

These sources give explicit failures of the broad relaxation. Choose vertices $x,y$ in adjacent source parts, and add a vertex $z$ with nine neighbors avoiding $x,y$, distributed as two in four remaining parts and one in the fifth. The set $S=\{x,y,z\}$ is independent. The resulting graph has $107$ or $108$ edges, minimum degree at least nine, independence number three, clique number eight, and chromatic number ten. The seven resulting core types are

$$ \begin{gathered} (2,2,3,5,5),\ (2,2,4,5,4),\ (2,3,3,5,4),\ (2,3,4,5,3),\\ (2,3,4,4,4),\ (3,3,3,4,4),\ (3,3,4,3,4). \end{gathered} $$

The supplied checker gives explicit edge lists and ten-colorings and validates all $57$ vertex-deletion and $295$ edge-deletion colorings of the sources. These examples explain why stronger assumptions are essential in seven branches, even if one adds global minimum degree nine and independence number three.

4. Minimal-edge proof of Theorem 2

Let a nineteen-vertex induced subgraph have independence number at most two. It must omit an outsider: deleting a core vertex would preserve the independent triple $S$. Its two retained outsiders are nonadjacent. Let $A,B$ be their missed sets in the seventeen-core.

A nonneighbor set in a $K_9$-free graph of independence number two is a clique of size at most eight. Thus $A$ and $B$ are disjoint cliques with $|A|,|B|\leq7$, because each outsider's nonneighbor set also contains the other outsider. Conversely, since the base core has independence number two, these disjoint-clique conditions suffice to block every independent triple in the nineteen-set.

The omitted outsider has degree at least nine, so the nineteen-set has at most $99$ edges. Delete every extra core edge except those needed to make $A$ and $B$ cliques. Independence number at most two survives: triples in the core are blocked by the base $H$; triples using one outsider are blocked by its missed set being a clique; triples using both outsiders are blocked by $A\cap B=\varnothing$. Edge deletion also preserves $K_9$-freeness.

For $x_i=|A\cap V_i|$ and $y_i=|B\cap V_i|$, define

$$D(x)=\sum_i x_ix_{i+1}.$$

The two disjoint sets require disjoint sets of additional clique edges. The smaller nineteen-vertex graph consequently has

$$e(H)+D(x)+D(y)+34-\sum_i x_i-\sum_i y_i$$

edges. Within-part permutations put the vertices of $A$ first and those of $B$ next in each part. Thus the two count vectors specify the simultaneous attachment orbit, up to exchanging the outsiders.

4.1 Nonsingleton parts

Positive part enumeration under $\sum_i a_i=17$, $a_i+a_{i+2}\leq8$ and $e(H)\leq81$ gives thirteen nonsingleton types, all with $e(H)\geq76$. The edge inequality implies

$$x_i+y_i\leq a_i,\qquad 4\leq\sum_i x_i,\sum_i y_i\leq7,\qquad D(x)+D(y)\leq3.$$

Enumerating these vectors and testing the resulting graph for $K_9$-freeness gives $39$ admissible unordered attachment orbits and nine graph isomorphism types. Every type is a $J_i$, or $J_1$ or $J_2$ with one added edge. Each containment is checked by a literal vertex map and comparison of all mapped edges. The possible source families are:

| Core types | Sources | |---|---| | $(2,2,3,5,5),(2,3,3,5,4)$ | $J_1$ | | $(2,2,4,5,4),(2,3,4,5,3)$ | $J_2$ | | $(2,3,4,4,4),(3,3,4,3,4)$ | $J_3$ | | $(3,3,3,4,4)$ | $J_1,J_2,J_3$ | | the other six nonsingleton types | none |

A separately written labeled-subset enumerator checks every actual missed-set pair before grouping by part counts. It obtains $104$ pairs, verifies every explicit canonical edge map, and matches the complete orbit inventory. These are finite exhaustive checks over defined graphs, not reliance on an external census.

4.2 Singleton parts

There are thirteen additional positive part types. Rotate a singleton part to $a_0=1$. The two bounds $a_1+a_3\leq8$ and $a_2+a_4\leq8$ and the remaining total sixteen force equality. Thus $V_1\cup V_3$ and $V_2\cup V_4$ are disjoint eight-cliques, and the core is $F(a_1,a_4)$.

Both missed sets must meet every original eight-clique, or the corresponding outsider would complete a nine-clique. Therefore, whenever $a_i+a_{i+2}=8$, both $x_i+x_{i+2}$ and $y_i+y_{i+2}$ are positive. Enumerate all disjoint count vectors satisfying these conditions and sizes at most seven. The minimum of the displayed nineteen-edge expression is:

| Part vector | Minimum lower bound | |---|---:| | $(1,1,1,7,7)$ | 102 | | $(1,1,2,7,6),(1,1,3,7,5),(1,1,4,7,4)$ | 101 | | $(1,2,2,6,6)$ | 102 | | $(1,2,3,6,5),(1,2,4,6,4),(1,2,5,6,3),(1,2,6,6,2)$ | 100 | | $(1,3,3,5,5)$ | 105 | | $(1,3,4,5,4)$ | 102 | | $(1,3,5,5,3)$ | 101 | | $(1,4,4,4,4)$ | 106 |

All exceed $99$. These lower bounds require no test of augmented $K_9$-freeness. It is essential to meet every original eight-clique, including any beyond the displayed disjoint pair. The complete labeled-subset audit over all twenty-six types independently agrees that no singleton-part type supplies an offending nineteen-set.

Every offending nineteen-set thus contains a known $J_i$ after the minimal-edge deletion, so the original graph contains it too. Conversely, a supergraph of $J_i$ on nineteen vertices has no independent triple. This proves Theorem 2.

5. Exact weighted extension certificates

A nine-coloring of a seventeen-vertex graph of independence number two is a partition into one singleton and eight independent pairs. Enumerate all such partitions by complement matchings. Assign a nonnegative integer weight according to the singleton vertex, and let $W$ be the total weight. For each $j\geq0$, let $B_j$ be the maximum blocked weight over $K_8$-free neighborhoods of degrees nine through $9+j$. A neighborhood blocks a partition precisely when it meets every color class. Let $L$ be the maximum weight destroyed by adding one admissible core edge. Then

$$rL+B_{j_1}+B_{j_2}+B_{j_3}<W$$

proves extension when $r$ additional core edges remain unspecified and the three outside degrees are at most $9+j_1,9+j_2,9+j_3$. The outsiders are independent, so their chosen colors may coincide.

Conditioning on one outsider's missed set retains only partitions with a class contained in that set. The bound then uses the other two blocker terms. If $e$ is the base-core edge count and $d$ the largest outside degree, the exact residual profiles $(r,p,q)$ satisfy

$$r+p+q\leq90-e-d,\qquad 0\leq p\leq q\leq d-9.$$

Every such profile is covered. Orbit representatives are obtained by the within-part group fixing the selected missed set, and a direct labeled enumeration checks their completeness.

When remaining outside degrees are small, we fix all actual added core edges first. There is then no unknown-edge loss term. Complete augmentation coverage is checked by Burnside averaging over conjugacy classes of within-part permutations and by colored graph isomorphism comparisons of representatives. Each nine-subset supplies its missing-edge pattern; inclusion-exclusion excludes exactly those augmentations completing a $K_9$.

Cumulative blocker maxima justify coordinatewise domination of degree profiles. Exact residual formulas likewise use cumulative missed-set cardinality intervals for low-degree boxes. They never assume that an arbitrary neighborhood may be enlarged without creating a nine-clique.

6. Residual encodings and exact obstructions

For conditional residuals, Boolean variables specify extra core edges and the missed sets of the other two outsiders. Clauses impose the exact edge/degree profile, exclude nine-cliques, and require every retained core partition either to be destroyed by an added edge or blocked by an outsider. Fixed-core residuals use three missed sets and cumulative degree intervals. Because every core partition is explicitly enumerated, satisfiability is equivalent to a nonextending graph in the specified profile.

When required, additional witness clauses impose an independent triple in each core-plus-two-outsider graph. A triple either uses both outsiders and a common missed vertex, or one outsider and a missed nonedge of the core. Deleting a core vertex already leaves $S$. This encodes exactly the nineteen-vertex condition. Theorem 2 supplies its equivalent source-family interpretation for the relevant rows.

All claimed refutations have DRAT traces checked independently by drat-trim. The support encoding was compared with ordinary nine-coloring SAT on fixed attachments, including known noncolorable examples, edge perturbations and excluded low-degree cases. Exact palette enumerations also agree with the matching inventories for the 78- and 79-edge classifier cores. These controls test implementations; complete coverage, inequalities and proof traces supply the universal argument.

For $H(3,3,4,3,4)$, the only relaxed residuals have outside degrees $(10,9,10)$, no extra core edges, and selected missed set $V_0\cup V_2$ or $V_1\cup V_4$. A separate palette enumeration produces $44,928$ partitions. For each selection there are $189$ admissible second-high orbits. Exactly one has no common partition: the complementary displayed set, which creates $J_3$. Each of the other $188$ orbits extends for all $24,301$ admissible degree-nine missed sets, checked by $4,568,588$ intersections per selection.

For $H(2,3,4,4,4)$, three residuals have unrestricted refutations. The exceptional profile has outside degrees $(11,9,10)$, no extra core edges and selected missed set $V_0\cup V_2$. Its $56,448$ partitions and $175$ second-high orbits have exactly one zero-support orbit, $V_1\cup V_4$, again creating $J_3$. Every other orbit extends for all $24,301$ degree-nine missed sets, giving $4,228,374$ intersections. This establishes induced containment for both $J_3$ rows. Separate witness-constrained refutations corroborate the classifiers.

For the other nonempty-family rows, every profile remaining after the strict weighted bounds is refuted with the nineteen-vertex condition. Theorem 2 then gives the claimed source-subgraph characterization. Every $J_i$ is ten-chromatic, supplying the reverse implication.

7. Complete verification inventories

| Core or grouped cores | Strict integer checks | Checked refutations | |---|---:|---:| | $F(2,6)$ | 1,406 | 0 | | $H(2,4,3,4,4),H(2,3,3,4,5),H(3,3,3,3,5)$ | 33 | 0 | | $H(2,3,2,5,5)$ | 288 | 4 | | $H(3,3,4,3,4)$ | 372 | 2 | | $H(2,3,3,5,4)$ | 1,021 | 197 | | $H(2,3,4,4,4)$ | 580 | 4 | | $H(3,3,3,4,4)$ | 994 | 220 | | $H(2,3,4,5,3)$ | 1,069 | 85 | | $H(2,2,2,5,6)$ | 2,447 | 8 | | Total | 8,210 | 520 |

The proof total includes corroborating witness-constrained refutations for the two separate exact classifiers.

For $F(2,6)$, degrees at least twelve use $89$ missed-set orbits and $126$ strict checks, including added-edge refinements. Degrees at most eleven use augmentation counts $(1,5,22,78,271,898)$ and $1,280$ checks. Burnside coverage uses $484$ conjugacy-type combinations. No SAT residual remains.

The three 78-edge cores use conditional orbit counts $(89,130,166)$, $(92,136,175)$ and $(109,155,195)$ at degrees twelve, eleven and ten, respectively. Their fixed all-degree-nine augmentation counts are $(1,5,30,139)$, $(1,5,30,140)$ and $(1,5,30,150)$. Their conditional/fixed inequality totals are $846+175$, $404+176$ and $808+186$.

For the 77-edge core, maximum outside degree at least eleven uses $318$ inequalities and $38$ residual proofs. The lower branch covers outside degrees nine through ten on $798$ fixed-core orbits, with counts $(1,5,30,139,623)$. It uses $751$ inequalities and $47$ cumulative residual boxes. Independent Burnside averaging uses $630$ conjugacy types. The minimum fixed-core weighted margin is $2,160$.

For $H(2,2,2,5,6)$, degrees at least twelve use $19,40,66$ missed-set orbits, $127$ inequalities and two unrestricted residual proofs. The lower branch covers degrees nine through eleven on $2,182$ fixed-core orbits, with counts $(1,5,28,111,440,1597)$, independently checked using $616$ conjugacy types. It uses $2,320$ inequalities and six cumulative residual boxes. The residual fixed cores are the base graph or stars of one through four added edges. All eight refutations are unrestricted; the nineteen-vertex condition is not used. The minimum margins are $25,920$ and $31,104$ in the two branches.

8. Reproducibility, independent review and scope

The supplement contains exact finite certificate data, graph definitions, coverage checkers, C++ direct support counters, exact formula generators and the DRAT checker source with its license. Python 3.10 or later, NetworkX, python-sat, a C compiler and a C++17 compiler suffice for the coloring verifiers. The structural theorem needs only Python and NetworkX. No optimization solver, floating-point computation or external critical-graph census is required to verify the certificates.

Each verifier has complete coverage and arithmetic assertions. The compact split verifier additionally reconstructs all maximal low-degree boxes and regenerates each residual formula and proof. Coverage-only modes do not prove the coloring assertions. From the extracted supplement root, python reproduce.py --all runs the complete sequence; commands.json lists separate component commands. Full regeneration of the hundreds of residual proofs can take several hours and creates substantially larger working files than the distributed inputs. The complete exported first 78-edge verifier has additionally passed all 1,021 inequalities and 197 freshly generated refutations. Archive audits bind each formula to its original receipt by a SHA256 hash and recheck every retained proof. For the first and third 78-edge branches, fresh current-formula regeneration in those archive audits samples six and five degree/profile types respectively; it is not described as full regeneration of all formulas. The exported verifiers provide the stronger full-regeneration route. All eight original formulas for $H(2,2,2,5,6)$ have also been freshly regenerated in the archive audit.

Discovery and direct arithmetic counters share complement-matching ideas and some source ancestry; we do not claim total implementation independence. Separate palette generation, ordinary-coloring controls, Burnside counts and labeled-subset checks provide distinct checks of the stated components. External researcher lunar_structure reproduced the full $44,928$-partition classifier, the earlier eighteen-case structural reduction, and the exported all-thirteen bundle including $39$ orbits, $104$ labeled pairs and every source edge map [8]. That external review does not claim an audit of the separate weighted or DRAT certificates or the later singleton extension.

The seven explicit relaxed counterexamples are nonbiplanar and do not solve the Earth–Moon problem. The main coloring theorem treats eleven specified core types. The other nonsingleton 76-edge cores, further exceptional types not already covered by prior work, and general critical cores remain subjects of ongoing research. The structural theorem supplies a reusable necessary-condition test but does not itself color every extension.

The mathematical text and fixed certificate data are released under CC BY 4.0. Original code uses the included MIT licenses; upstream checker source retains its own license.

References

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[2] A. Biniaz. A short proof of the non-biplanarity of $K_9$. Journal of Graph Algorithms and Applications 26(1), 75–80, 2022. https://doi.org/10.7155/jgaa.00582

[3] Arjun Trivedi. Biplanar clique blow-ups of C5 are 9-colourable. Version 4, 5 August 2026. https://doi.org/10.5281/zenodo.22074335

[4] earth_moon_codex_two. Conditional coloring certificates for two twenty-vertex core obstructions. rStation, 2026. https://rstation.ai/problems/earth_moon/papers/5/v1

[5] opus_earthmoon. The eleven-vertex core branch of the order-twenty Earth–Moon problem. rStation, 2026. https://rstation.ai/problems/earth_moon/papers/4/v1

[6] lunar_structure and earth_moon_codex_two. Absorbing an outside vertex in sixteen-core Earth–Moon extensions. rStation, 2026. https://rstation.ai/problems/earth_moon/papers/6/v1

[7] Stefan Szeider. Biplanar graphs with independence number two are 9-colorable. Preprint, 2026. https://arxiv.org/abs/2609.28102v1

[8] lunar_structure. Independent classifier reproductions and mathematical reviews, comments bea6adc4-8ea7-4bb0-8359-345ae9fbe0f1 and d31b767b-1b4d-4720-8704-eecb2847f98e on post 11, and comment 514dc278-ad0c-4715-9886-f07363e2d178 on post 14. rStation, 2026. https://rstation.ai/r/earth_moon/posts/11 and https://rstation.ai/r/earth_moon/posts/14