diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux index 024f1e4..e9edb24 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.aux @@ -17,4 +17,5 @@ \@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} diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk index 7e53d2b..d1f118a 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"] 1778732745 "paper.tex" "paper.pdf" "paper" 1778732746 +["pdflatex"] 1778733012 "paper.tex" "paper.pdf" "paper" 1778733012 "/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 "" @@ -122,8 +122,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1778732746 1438 b028da99404765220b31b8b219d41cca "pdflatex" - "paper.tex" 1778732735 12434 2966cec5ecd6b41acdbce2ab2207a710 "" + "paper.aux" 1778733012 1496 56d68f0294258d5724b6317ee89e6aaa "pdflatex" + "paper.tex" 1778733006 13839 954cc3cf73e27b836fc30fd8b0d7a40b "" (generated) "paper.aux" "paper.log" diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.log b/papers/flip_symmetric_maximal_planar_graphs/paper.log index 13c28e1..7b97ae0 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.log +++ b/papers/flip_symmetric_maximal_planar_graphs/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) 14 MAY 2026 00:25 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 00:30 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -504,10 +504,10 @@ hs \OT1/cmr/m/n/10 with \OT1/cmtt/m/n/10 minimum[]connectivity $\OT1/cmr/m/n/10 [2] [3] [4] (./paper.aux) ) Here is how much of TeX's memory you used: - 13155 strings out of 478268 - 265572 string characters out of 5846347 - 542781 words of memory out of 5000000 - 30993 multiletter control sequences out of 15000+600000 + 13156 strings out of 478268 + 265610 string characters out of 5846347 + 542791 words of memory out of 5000000 + 30994 multiletter control sequences out of 15000+600000 475834 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -526,7 +526,7 @@ mti10.pfb> -Output written on paper.pdf (4 pages, 186074 bytes). +Output written on paper.pdf (4 pages, 189743 bytes). PDF statistics: 90 PDF objects out of 1000 (max. 8388607) 55 compressed objects within 1 object stream diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.pdf b/papers/flip_symmetric_maximal_planar_graphs/paper.pdf index 97ef191..30a5972 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 2e5f98c..1aa4b4c 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.tex +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.tex @@ -318,6 +318,38 @@ 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} %-----------------------------------------------------------------------