Refute min-degree-5 plane diamond coloring conjecture at order 28
Adds search_min_degree_counterexample_comprehensive iterating Sage's planar_graphs generator with minimum_degree=5. Exhaustive enumeration through order 27 (456,967 maximal planar graphs of minimum degree at least 5) finds no counterexample to Conjecture 2.4. At order 28, three counterexamples are exhibited and verified via Sage's chromatic_number on the auxiliary graph, refuting the conjecture. Updates paper with the refutation theorem, the per-order census, a figure of one counterexample, and graph6 strings of the other two. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\newlabel{def:diamond}{{2.3}{2}}
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\newlabel{thm:counterexample}{{3.3}{2}}
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\bibcite{appel1977every}{1}
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\bibcite{robertson1997four}{2}
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\bibcite{appel1977every}{1}
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\bibcite{robertson1997four}{2}
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Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring.
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\end{conjecture}
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\begin{remark}
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We have verified Conjecture~\ref{conj:mindeg5} computationally for all maximal planar graphs of minimum degree at least $5$ and order at most $N$, by exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}. No counterexample has been found.
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\end{remark}
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\begin{theorem} \label{thm:mindeg5counterexample}
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Conjecture~\ref{conj:mindeg5} is false. The smallest counterexamples have order $28$, and every maximal planar graph of minimum degree at least $5$ and order at most $27$ admits a plane diamond coloring.
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\end{theorem}
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\begin{proof}
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By exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} generator (with \texttt{minimum\_connectivity=3}, \texttt{maximum\_face\_size=3}, and \texttt{minimum\_degree=5}) and the auxiliary-graph reduction described in the proof of Theorem~\ref{thm:counterexample}, every maximal planar graph of minimum degree at least $5$ and order in $\{12, 13, \dots, 27\}$ admits a plane diamond coloring. The numbers of such triangulations at orders $12, 13, \dots, 27$ are
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\[
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1,\ 0,\ 1,\ 1,\ 3,\ 4,\ 12,\ 23,\ 73,\ 192,\ 651,\ 2070,\ 7290,\ 25381,\ 91441,\ 329824,
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\]
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totalling $456{,}967$ graphs, none of which is a counterexample.
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At order $28$, however, counterexamples do exist. The graph in Figure~\ref{fig:mindeg5counterexample} is one such, with canonical graph6 string
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\[
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\verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??B???|C?CIG?GIA?iD@?TPC?VQG_Bi+.
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\]
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It has $|V| = 28$, $|E| = 78 = 3 \cdot 28 - 6$, minimum degree $5$, and chromatic number $4$. Two further counterexamples at order $28$ have canonical graph6 strings
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\[
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\verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+
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\]
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and
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\[
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\verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+ .
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\]
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Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $\chi(H_u) > 4$ for every $u$ in each of these graphs.
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\end{proof}
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\begin{figure}[htbp]
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\centering
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\includegraphics[width=0.45\textwidth]{min_degree_5_counterexample.png}
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\caption{One of three known smallest counterexamples to Conjecture~\ref{conj:mindeg5}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}
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\label{fig:mindeg5counterexample}
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\end{figure}
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\begin{thebibliography}{9}
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