From 2ae729db1e9bafdb8399165d237a70cbb6988448 Mon Sep 17 00:00:00 2001 From: didericis Date: Sat, 9 May 2026 12:40:42 -0400 Subject: [PATCH] Refute plane diamond coloring conjecture with 16-vertex counterexample Adds Theorem 2.6 stating Conjecture 2.5 is false, with proof exhibiting a 16-vertex maximal planar graph (graph6 O???IAGKEBEQQYHdplW{n) for which the auxiliary 4-colorability check fails at every root vertex, verified computationally via Sage's chromatic_number. Includes the graph as a figure and adds a McKay graph6 reference. Co-Authored-By: Claude Opus 4.7 --- .../plane_diamond_coloring/counterexample.png | Bin 0 -> 38090 bytes papers/plane_diamond_coloring/paper.aux | 8 +- .../plane_diamond_coloring/paper.fdb_latexmk | 21 ++- papers/plane_diamond_coloring/paper.fls | 93 ++++++++++ papers/plane_diamond_coloring/paper.log | 166 +++++++++++++----- papers/plane_diamond_coloring/paper.pdf | Bin 176410 -> 236597 bytes papers/plane_diamond_coloring/paper.tex | 41 +++++ 7 files changed, 277 insertions(+), 52 deletions(-) create mode 100755 papers/plane_diamond_coloring/counterexample.png diff --git a/papers/plane_diamond_coloring/counterexample.png b/papers/plane_diamond_coloring/counterexample.png new file mode 100755 index 0000000000000000000000000000000000000000..ec373784455dcc770d5014e0b270e88ccb7ca3ca GIT binary patch literal 38090 zcmaI8c|4Wd`vzP}2^AtkQb?KSA#<^}d1$jYG7rg2qRf;`88eG0+q@CVlraiPl3^ob z5kg3jnfHEl&i6gP_mB6FKAlhJdDe5UweEEf*LAJ^@S2wLadIZ|0|yQqS5-mj95`^0 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b/papers/plane_diamond_coloring/paper.aux @@ -4,12 +4,16 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Results}}{1}{}\protected@file@percent } \citation{appel1977every} \citation{robertson1997four} +\citation{mckaygraph6} \bibcite{appel1977every}{1} \bibcite{robertson1997four}{2} +\bibcite{mckaygraph6}{3} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{2}{}\protected@file@percent } -\gdef \@abspage@last{2} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A maximal planar graph on $16$ vertices with no plane diamond coloring.}}{3}{}\protected@file@percent } +\newlabel{fig:counterexample}{{1}{3}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } +\gdef \@abspage@last{3} diff --git a/papers/plane_diamond_coloring/paper.fdb_latexmk b/papers/plane_diamond_coloring/paper.fdb_latexmk index 19e2474..35d370d 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"] 1778344156 "paper.tex" "paper.pdf" "paper" 1778344156 +["pdflatex"] 1778344761 "paper.tex" "paper.pdf" "paper" 1778344761 "/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 "" @@ -20,6 +20,8 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti7.tfm" 1136768653 1492 86331993fe614793f5e7e755835c31c5 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt8.tfm" 1136768653 768 d7b9a2629a0c353102ad947dc9221d49 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" @@ -33,7 +35,10 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti7.pfb" 1248133631 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zdW?J4xLD*728#^a+LWRpsIqxp;*MXFz@X%1oa(zTK#uNZw4T+$QG7C_+m4uywf}f< zh8x|r7~=ke6*;pSFV8SLW=`hE7FW^NsnHj=h7f6I&`W|kID;&FLr!tqLlZ?SHzNIM_@Wm^@owq{XkLM7=(GDw6_c-`iAWzDA(Jcz924MSGk{}Ov+=bc z;bDVg)*{hkC1EFF{nAvOoZLycc}ZAERNI0X3;lQD~8hp`g$+&rOq&;F7_8eLm{NL?efT4SaXVtV<w`q?QnH%edb{qRR&q{p2jS8 zx?Wjy-(3wj*MnX7Ij-TPPNq*i-j|3T_ZG)0l|MmscxZ^>EF8@ 4$ for every $u \in V(G)$, so $G$ admits no plane diamond coloring. +\end{proof} + +\begin{figure}[htbp] + \centering + \includegraphics[width=0.45\textwidth]{counterexample.png} + \caption{A maximal planar graph on $16$ vertices with no plane diamond coloring.} + \label{fig:counterexample} +\end{figure} + \begin{thebibliography}{9} \bibitem{appel1977every} @@ -159,6 +195,11 @@ N.~Robertson, D.~Sanders, P.~Seymour, and R.~Thomas, \emph{The four-colour theorem}, Journal of Combinatorial Theory, Series B, vol.~70, no.~1, pp.~2--44, 1997. +\bibitem{mckaygraph6} +B.~D.~McKay, +\emph{Description of graph6, sparse6 and digraph6 encodings}, +\url{https://users.cecs.anu.edu.au/~bdm/data/formats.txt}. + \end{thebibliography} \end{document}