diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux index 0b58443..024f1e4 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.aux @@ -8,10 +8,13 @@ \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 {}{6}{Further necessary properties of a minimal counterexample}}{3}{}\protected@file@percent } +\@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}} \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 56970d7..7e53d2b 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"] 1778732276 "paper.tex" "paper.pdf" "paper" 1778732276 +["pdflatex"] 1778732745 "paper.tex" "paper.pdf" "paper" 1778732746 "/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 "" @@ -29,9 +29,11 @@ "/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 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 "" "/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/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" "/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcore.code.tex" 1601326656 992 855ff26741653ab54814101ca36e153c "" "/usr/local/texlive/2022/texmf-dist/tex/generic/pgf/basiclayer/pgfcorearrows.code.tex" 1601326656 43820 1fef971b75380574ab35a0d37fd92608 "" @@ -120,8 +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" 1778732276 1234 809d17d72706986551d4267509a33a08 "pdflatex" - "paper.tex" 1778732267 11154 4367545e63b37b360cb8c3c1c287d046 "" + "paper.aux" 1778732746 1438 b028da99404765220b31b8b219d41cca "pdflatex" + "paper.tex" 1778732735 12434 2966cec5ecd6b41acdbce2ab2207a710 "" (generated) "paper.aux" "paper.log" diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.fls b/papers/flip_symmetric_maximal_planar_graphs/paper.fls index 1b2b4cb..94c6303 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.fls +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.fls @@ -445,6 +445,8 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pf INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.log b/papers/flip_symmetric_maximal_planar_graphs/paper.log index 1530de5..13c28e1 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 00:17 +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 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: - 13153 strings out of 478268 - 265523 string characters out of 5846347 - 542761 words of memory out of 5000000 - 30991 multiletter control sequences out of 15000+600000 + 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 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 @@ -521,13 +521,15 @@ sfonts/cm/cmr10.pfb> -Output written on paper.pdf (4 pages, 165847 bytes). +/cmsy7.pfb> +Output written on paper.pdf (4 pages, 186074 bytes). PDF statistics: - 80 PDF objects out of 1000 (max. 8388607) - 49 compressed objects within 1 object stream + 90 PDF objects out of 1000 (max. 8388607) + 55 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 d962ad4..97ef191 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 34fac10..2e5f98c 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.tex +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.tex @@ -289,6 +289,35 @@ 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} + \end{document} %-----------------------------------------------------------------------