diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux index e9edb24..b17b3d3 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.aux @@ -8,14 +8,9 @@ \newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.1}{2}} \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Flip symmetry frequency}}{2}{}\protected@file@percent } \newlabel{sec:frequency}{{5}{2}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Further necessary properties of a minimal counterexample}}{3}{}\protected@file@percent } \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Edge-deletion subgraphs}}{4}{}\protected@file@percent } -\newlabel{def:edge-deletion}{{7.1}{4}} -\newlabel{thm:edge-deletion-4colorable}{{7.2}{4}} -\newlabel{thm:edge-deletion-coloring-structure}{{7.3}{4}} -\gdef \@abspage@last{4} +\gdef \@abspage@last{3} diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk index d1f118a..8febd69 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778733012 "paper.tex" "paper.pdf" "paper" 1778733012 +["pdflatex"] 1778734485 "paper.tex" "paper.pdf" "paper" 1778734486 "/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 "" @@ -29,11 +29,9 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" - 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PDF statistics: - 90 PDF objects out of 1000 (max. 8388607) - 55 compressed objects within 1 object stream + 77 PDF objects out of 1000 (max. 8388607) + 47 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.pdf b/papers/flip_symmetric_maximal_planar_graphs/paper.pdf index 30a5972..072cd63 100644 Binary files a/papers/flip_symmetric_maximal_planar_graphs/paper.pdf and b/papers/flip_symmetric_maximal_planar_graphs/paper.pdf differ diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.tex b/papers/flip_symmetric_maximal_planar_graphs/paper.tex index 1aa4b4c..89a49e5 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.tex +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.tex @@ -276,80 +276,6 @@ the minimum-degree-$5$ class --- which already contains every candidate minimum-order $5$-chromatic graph --- flip-symmetric examples become a vanishing fraction. -\section{Further necessary properties of a minimal counterexample} - -The frequency data of Section~\ref{sec:frequency} look unflattering -only when flip-symmetry is weighed against the full class of maximal -planar graphs. The class that actually matters --- minimum-order -$5$-chromatic triangulations that also resist every Kempe-style -reduction --- is far thinner, and flip-symmetry may exclude a -substantially larger fraction of it if the configurations it removes -overlap those responsible for Kempe reducibility. We therefore turn -to identifying further necessary properties of a minimum-order -$5$-chromatic maximal planar graph, of which flip-asymmetry is the -first. - -\section{Edge-deletion subgraphs} - -\begin{definition}[Edge-deletion subgraph]\label{def:edge-deletion} -Let $G$ be a maximal planar graph and $uv \in E(G)$. The -\emph{edge-deletion subgraph at $uv$} is the spanning subgraph -$G - uv = (V(G),\,E(G) \setminus \{uv\})$. Write -$\mathcal{D}(G) = \{G - uv : uv \in E(G)\}$. -\end{definition} - -\begin{theorem}\label{thm:edge-deletion-4colorable} -Let $G_0$ be a maximal planar graph of minimum order with -$\chi(G_0) \geq 5$. Then every $H \in \mathcal{D}(G_0)$ is -$4$-colorable. -\end{theorem} - -\begin{proof} -Fix $uv \in E(G_0)$ and let $G_0 / uv$ denote the simple planar graph -obtained by contracting $uv$ and discarding parallel edges. Since -$|V(G_0/uv)| = |V(G_0)| - 1$, the minimality of $G_0$ supplies a -proper $4$-coloring $c$ of $G_0 / uv$. Let $z$ be the contracted -vertex and define $c'\colon V(G_0) \to \{1,2,3,4\}$ by -$c'(u) = c'(v) = c(z)$ and $c'(y) = c(y)$ for $y \notin \{u, v\}$. -Every edge of $G_0 - uv$ is either disjoint from $\{u, v\}$ or -incident to exactly one of them; in either case the corresponding -edge of $G_0 / uv$ has distinct endpoints under $c$, so $c'$ assigns -its endpoints distinct colors. The edge $uv$ itself is absent from -$G_0 - uv$, so $c'$ is a proper $4$-coloring of $G_0 - uv$. -\end{proof} - -\begin{theorem}\label{thm:edge-deletion-coloring-structure} -Let $G_0$ be a maximal planar graph of minimum order with -$\chi(G_0) \geq 5$, fix $uv \in E(G_0)$, and let $\varphi$ be any -proper $4$-coloring of $G_0 - uv$. Write $a = \varphi(u)$ and let -$b, c, d$ denote the three remaining colors. Then: -\begin{enumerate} -\item $\varphi(v) = a$; -\item the subgraph of $G_0 - uv$ induced by the vertices of color -$a$ or $b$ contains a path from $u$ to $v$; -\item the subgraph of $G_0 - uv$ induced by the vertices of color -$a$ or $c$ contains a path from $u$ to $v$. -\end{enumerate} -\end{theorem} - -\begin{proof} -(1) If $\varphi(v) \neq a$ then $\varphi$ is already a proper -$4$-coloring of $G_0$, since the only edge of $G_0$ absent from -$G_0 - uv$ is $uv$ and its endpoints have distinct colors. This -contradicts $\chi(G_0) \geq 5$, so $\varphi(v) = a$. - -(2) Suppose, for contradiction, that $u$ and $v$ lie in distinct -connected components of the subgraph of $G_0 - uv$ induced by the -color classes $a$ and $b$. Let $C$ be the component containing $u$, -and define $\varphi'\colon V(G_0) \to \{a,b,c,d\}$ by swapping colors -$a \leftrightarrow b$ on $C$ and leaving every other vertex -unchanged. Then $\varphi'$ is a proper $4$-coloring of $G_0 - uv$ -with $\varphi'(u) = b$ and $\varphi'(v) = a$, contradicting part~(1) -applied to $\varphi'$. - -(3) Identical to (2) with $c$ in place of $b$. -\end{proof} - \end{document} %-----------------------------------------------------------------------