diff --git a/papers/plane_diamond_coloring/min_degree_5_counterexample.png b/papers/plane_diamond_coloring/min_degree_5_counterexample.png new file mode 100755 index 0000000..df1996e Binary files /dev/null and b/papers/plane_diamond_coloring/min_degree_5_counterexample.png differ diff --git a/papers/plane_diamond_coloring/paper.aux b/papers/plane_diamond_coloring/paper.aux index 42f2f6a..aae24a5 100644 --- a/papers/plane_diamond_coloring/paper.aux +++ b/papers/plane_diamond_coloring/paper.aux @@ -12,16 +12,19 @@ \newlabel{def:diamond}{{2.3}{2}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Results}}{2}{}\protected@file@percent } \newlabel{thm:counterexample}{{3.3}{2}} -\bibcite{appel1977every}{1} -\bibcite{robertson1997four}{2} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The unique smallest maximal planar graph with no plane diamond coloring; it has $13$ vertices and degree sequence $(6,6,6,6,6,6,6,5,5,4,4,3,3)$.}}{3}{}\protected@file@percent } \newlabel{fig:counterexample}{{1}{3}} \newlabel{conj:mindeg5}{{3.4}{3}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{3}{}\protected@file@percent } +\newlabel{thm:mindeg5counterexample}{{3.5}{3}} +\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{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 } +\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 71fadc2..8455449 100644 --- a/papers/plane_diamond_coloring/paper.fdb_latexmk +++ b/papers/plane_diamond_coloring/paper.fdb_latexmk @@ -1,6 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778347081 "/Users/didericis/Code/math-research/papers/plane_diamond_coloring/paper.tex" "paper.pdf" "paper" 1778347082 - 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PDF statistics: - 95 PDF objects out of 1000 (max. 8388607) + 97 PDF objects out of 1000 (max. 8388607) 56 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) - 6 words of extra memory for PDF output out of 10000 (max. 10000000) + 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/plane_diamond_coloring/paper.pdf b/papers/plane_diamond_coloring/paper.pdf index 7177dcb..f828187 100644 Binary files a/papers/plane_diamond_coloring/paper.pdf and b/papers/plane_diamond_coloring/paper.pdf differ diff --git a/papers/plane_diamond_coloring/paper.tex b/papers/plane_diamond_coloring/paper.tex index f999729..9faa75d 100644 --- a/papers/plane_diamond_coloring/paper.tex +++ b/papers/plane_diamond_coloring/paper.tex @@ -164,9 +164,38 @@ For minimality and uniqueness, we exhaustively enumerated every maximal planar g Every maximal planar graph $G$ of minimum degree at least $5$ has a plane diamond coloring. \end{conjecture} -\begin{remark} -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. -\end{remark} +\begin{theorem} \label{thm:mindeg5counterexample} +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. +\end{theorem} + +\begin{proof} +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 +\[ + 1,\ 0,\ 1,\ 1,\ 3,\ 4,\ 12,\ 23,\ 73,\ 192,\ 651,\ 2070,\ 7290,\ 25381,\ 91441,\ 329824, +\] +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 +\[ + \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??B???|C?CIG?GIA?iD@?TPC?VQG_Bi+. +\] +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 +\[ + \verb+[?`???I@PCAG????@COGaGA_OD?DD?Aa_AII?PPV???Y??@ii?ATT?@T?T@agAgX+ +\] +and +\[ + \verb+[??DAaGP@OA_AI@DCPOaI_gh@PO?????C??BIA??gG?PC?IPC?Ig_?tIG?TO??F~+ . +\] +Direct computation (using \texttt{Sage}'s \texttt{chromatic\_number}) verifies $\chi(H_u) > 4$ for every $u$ in each of these graphs. +\end{proof} + +\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.} + \label{fig:mindeg5counterexample} +\end{figure} \begin{thebibliography}{9}