From 256cbbf90a327721239e15075672cfb85c7559cb Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 9 May 2026 16:16:58 -0400 Subject: [PATCH] Complete n=28 census: 4 counterexamples among 1,204,737 min-deg-5 triangulations 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 --- papers/plane_diamond_coloring/paper.aux | 2 +- .../plane_diamond_coloring/paper.fdb_latexmk | 6 +++--- papers/plane_diamond_coloring/paper.log | 6 +++--- papers/plane_diamond_coloring/paper.pdf | Bin 268316 -> 268670 bytes papers/plane_diamond_coloring/paper.tex | 14 ++++++++------ 5 files changed, 15 insertions(+), 13 deletions(-) diff --git a/papers/plane_diamond_coloring/paper.aux b/papers/plane_diamond_coloring/paper.aux index aae24a5..38deb64 100644 --- a/papers/plane_diamond_coloring/paper.aux +++ b/papers/plane_diamond_coloring/paper.aux @@ -24,7 +24,7 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@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} diff --git a/papers/plane_diamond_coloring/paper.fdb_latexmk b/papers/plane_diamond_coloring/paper.fdb_latexmk index 8455449..873e00e 100644 --- a/papers/plane_diamond_coloring/paper.fdb_latexmk +++ b/papers/plane_diamond_coloring/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778355822 "paper.tex" "paper.pdf" "paper" 1778355822 +["pdflatex"] 1778357729 "paper.tex" "paper.pdf" "paper" 1778357729 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -62,8 +62,8 @@ "/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" diff --git a/papers/plane_diamond_coloring/paper.log b/papers/plane_diamond_coloring/paper.log index 984ab59..253e17a 100644 --- a/papers/plane_diamond_coloring/paper.log +++ b/papers/plane_diamond_coloring/paper.log @@ -1,4 +1,4 @@ -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 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -204,7 +204,7 @@ Package pdftex.def Info: counterexample.png used on input line 158. 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z4=e)pjsJ1q0}##i5bUh1l&q|<-^HYb;n%(bdEHa9LS=NcS2nVR)Bw10vL>%&xKem* zq6@Ed_%Lt;j%jkd_TD@yNLIT21ATJBO<6o50X-$X$?vs~yS78nmrYCXkR1eB2hTz1|BIJq})=&7jrgJaa+xZN|`tc$8) zGJ71Vzc_asyab&-4sR2iI}RFQYV!F!Dq-MXyA-So@U+mWq`I@$WJCAWnp5r-X9Nng z{2e|Q&aENmT8P(z=M;T%?YtUZN!k(Hc_;^UA5i>4jUP?x>5J|b#N={5lYdirMT{g8 p6#Czte0KOh$;sRf!_ED(tGk!0l{E&hfS{m&AO-`299SOXe*iQh+!p`< diff --git a/papers/plane_diamond_coloring/paper.tex b/papers/plane_diamond_coloring/paper.tex index 9faa75d..8631c68 100644 --- a/papers/plane_diamond_coloring/paper.tex +++ b/papers/plane_diamond_coloring/paper.tex @@ -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}