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Complete n=28 census: 4 counterexamples among 1,204,737 min-deg-5 triangulations

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Exhaustive enumeration at order 28 finished with exactly four maximal
planar graphs of minimum degree 5 lacking a plane diamond coloring,
out of 1,204,737 total. Adds the fourth counterexample's canonical
graph6 string and updates the figure caption.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-09 16:16:58 -04:00
parent 4f2a703c12
commit 256cbbf90a
5 changed files with 15 additions and 13 deletions
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papers/plane_diamond_coloring/paper.aux
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\newlabel{tocindent1}{17.77782pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces One of three known smallest counterexamples to Conjecture\nonbreakingspace 3.4\hbox {}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}}{4}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces One of four smallest counterexamples to Conjecture\nonbreakingspace 3.4\hbox {}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}}{4}{}\protected@file@percent }
\newlabel{fig:mindeg5counterexample}{{2}{4}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{4}{}\protected@file@percent }
\gdef \@abspage@last{4}
papers/plane_diamond_coloring/paper.fdb_latexmk
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# Fdb version 3
["pdflatex"] 1778355822 "paper.tex" "paper.pdf" "paper" 1778355822
["pdflatex"] 1778357729 "paper.tex" "paper.pdf" "paper" 1778357729
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"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
"counterexample.png" 1778345471 31680 49acd944c061e194d20aac1d5cd8ef86 ""
"min_degree_5_counterexample.png" 1778355623 56300 15b8c2d55f1dc6b52f306af893b9fd60 ""
"paper.aux" 1778355822 1755 52c9cc0d4e80337d3db689115d06b369 "pdflatex"
"paper.tex" 1778355812 12440 9687ca443e7a03568da7d185c37d3adf ""
"paper.aux" 1778357729 1748 2280164128caceebcf749b9b57af5b08 "pdflatex"
"paper.tex" 1778357722 12657 98329ea51ea0ea48c39cd2520176c070 ""
(generated)
"paper.aux"
"paper.log"
papers/plane_diamond_coloring/paper.log
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 15:43
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 9 MAY 2026 16:15
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@@ -204,7 +204,7 @@ Package pdftex.def Info: counterexample.png used on input line 158.
File: min_degree_5_counterexample.png Graphic file (type png)
<use min_degree_5_counterexample.png>
Package pdftex.def Info: min_degree_5_counterexample.png used on input line 19
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[4 <./min_degree_5_counterexample.png>]
(./paper.aux) )
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Output written on paper.pdf (4 pages, 268670 bytes).
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papers/plane_diamond_coloring/paper.tex
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@@ -175,17 +175,19 @@ By exhaustive enumeration via \texttt{Sage}'s \texttt{graphs.planar\_graphs} gen
\]
totalling $456{,}967$ graphs, none of which is a counterexample.
At order $28$, however, counterexamples do exist. The graph in Figure~\ref{fig:mindeg5counterexample} is one such, with canonical graph6 string
At order $28$, however, counterexamples do exist. Of the $1{,}204{,}737$ maximal planar graphs of minimum degree at least $5$ and order $28$, exactly four lack a plane diamond coloring. The graph in Figure~\ref{fig:mindeg5counterexample} is one such, with canonical graph6 string
\[
\verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??B???|C?CIG?GIA?iD@?TPC?VQG_Bi+.
\]
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
It has $|V| = 28$, $|E| = 78 = 3 \cdot 28 - 6$, minimum degree $5$, and chromatic number $4$. The remaining three counterexamples at order $28$ have canonical graph6 strings
\[
\verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+
\verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+,
\]
and
\[
\verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+ .
\verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+,
\]
\[
\verb+[C_OQ?_?O@?a?aOOCC??A??GCCOCCO?gg?II?SSO@PPI_I_I_I_}??@yi_?LTS?B+ .
\]
Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $\chi(H_u) > 4$ for every $u$ in each of these graphs.
\end{proof}
@@ -193,7 +195,7 @@ Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{min_degree_5_counterexample.png}
\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.}
\caption{One of four smallest counterexamples to Conjecture~\ref{conj:mindeg5}: a maximal planar graph on $28$ vertices with minimum degree $5$ admitting no plane diamond coloring.}
\label{fig:mindeg5counterexample}
\end{figure}