diff --git a/papers/colored_edge_flip_classes/paper.aux b/papers/colored_edge_flip_classes/paper.aux index ceee268..0524451 100644 --- a/papers/colored_edge_flip_classes/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -6,9 +6,9 @@ \newlabel{def:flip-symmetric}{{3.1}{2}} \newlabel{def:flip-neighborhood}{{3.2}{2}} \newlabel{def:colored-flip-class}{{3.3}{2}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The flip neighborhood of a minimum-order counterexample}}{2}{}\protected@file@percent } -\newlabel{def:edge-deletion}{{4.1}{2}} -\newlabel{lem:edge-deletion-4colorable}{{4.2}{2}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The flip neighborhood of a minimum-order counterexample}}{3}{}\protected@file@percent } +\newlabel{def:edge-deletion}{{4.1}{3}} +\newlabel{lem:edge-deletion-4colorable}{{4.2}{3}} \newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}} \newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}} \newlabel{tocindent-1}{0pt} diff --git a/papers/colored_edge_flip_classes/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk index 5fff4d7..408afa8 100644 --- a/papers/colored_edge_flip_classes/paper.fdb_latexmk +++ b/papers/colored_edge_flip_classes/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778743139 "paper.tex" "paper.pdf" "paper" 1778743140 +["pdflatex"] 1778743194 "paper.tex" "paper.pdf" "paper" 1778743195 "/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 "" @@ -29,13 +29,16 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb" 1248133631 36094 798f80770b3b148ceedd006d487db67c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb" 1248133631 32734 69e00a6b65cedb993666e42eedb3d48f "" "/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/cmr9.pfb" 1248133631 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" "/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/cmsy8.pfb" 1248133631 32626 4f5c1b83753b1dd3a97d1b399a005b4b "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy9.pfb" 1248133631 32442 c975af247b6702f7ca0c299af3616b80 "" "/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 "" @@ -128,8 +131,8 @@ "/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 "" - 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PDF statistics: - 105 PDF objects out of 1000 (max. 8388607) - 64 compressed objects within 1 object stream + 120 PDF objects out of 1000 (max. 8388607) + 73 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/colored_edge_flip_classes/paper.pdf b/papers/colored_edge_flip_classes/paper.pdf index 4cd66f6..9c835cf 100644 Binary files a/papers/colored_edge_flip_classes/paper.pdf and b/papers/colored_edge_flip_classes/paper.pdf differ diff --git a/papers/colored_edge_flip_classes/paper.tex b/papers/colored_edge_flip_classes/paper.tex index e938fa2..b837740 100644 --- a/papers/colored_edge_flip_classes/paper.tex +++ b/papers/colored_edge_flip_classes/paper.tex @@ -68,13 +68,25 @@ \subjclass[2010]{Primary } -\keywords{} +\keywords{Four Color Theorem, edge flip, maximal planar graph, +Kempe chain, minimum counterexample} \date{} \dedicatory{} \begin{abstract} +Suppose the Four Color Theorem fails, and let $G_0$ be a maximal +planar graph of minimum order with $\chi(G_0) \geq 5$. Using an +edge-deletion argument together with a Kempe-chain swap, we show +that every graph in the flip neighborhood $\mathcal{N}(G_0)$ --- the +set of maximal planar graphs obtainable from $G_0$ by a single +admissible edge flip --- is $4$-colorable. In particular, no such +$G_0$ is flip-symmetric. We also introduce the colored edge flip +class $\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a +proper $4$-coloring $\varphi$ of $H$, and record that +$G_0 \notin \mathcal{C}(H, \varphi)$ for any +$H \in \mathcal{N}(G_0)$ and any such $\varphi$. \end{abstract} \maketitle