From 31dd21786311184638e97826ae0c076d43f765f1 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 03:17:03 -0400 Subject: [PATCH] Justify non-triangulated step in Lemma 4.2 contraction proof The original proof appealed to minimality of $G_0$ to 4-color $G_0/uv$, but $G_0/uv$ is not in general a triangulation, so it is not directly covered by the minimality hypothesis (which is over maximal planar graphs). Triangulate $G_0/uv$ into a maximal planar $T$ on the same vertex set: $|V(T)| < |V(G_0)|$, so minimality gives $T$ a 4-coloring, which restricts to $G_0/uv$. Co-Authored-By: Claude Opus 4.7 --- papers/colored_edge_flip_classes/paper.aux | 6 +-- .../paper.fdb_latexmk | 7 +-- papers/colored_edge_flip_classes/paper.fls | 1 + papers/colored_edge_flip_classes/paper.log | 46 ++++++++---------- papers/colored_edge_flip_classes/paper.pdf | Bin 211837 -> 212549 bytes papers/colored_edge_flip_classes/paper.tex | 11 +++-- 6 files changed, 37 insertions(+), 34 deletions(-) diff --git a/papers/colored_edge_flip_classes/paper.aux b/papers/colored_edge_flip_classes/paper.aux index 8560cc4..d05375a 100644 --- a/papers/colored_edge_flip_classes/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -9,14 +9,14 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{A minimal four-colorable counterexample}}{2}{}\protected@file@percent } \newlabel{def:edge-deletion}{{4.1}{2}} \newlabel{lem:edge-deletion-4colorable}{{4.2}{2}} -\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{2}} +\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}} +\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}} +\newlabel{thm:no-colored-class-contains-G}{{4.5}{3}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}} -\newlabel{thm:no-colored-class-contains-G}{{4.5}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Case\nonbreakingspace 2 of the proof of Theorem\nonbreakingspace 4.4\hbox {}: $u, v$ share color $a$ and $w, x$ share color $c$. The $\{a, b\}$-Kempe path $P$ from $u$ to $v$ separates $w$ from $x$ in the plane, so no $\{c, d\}$-path between $w$ and $x$ can avoid crossing $P$; since the color sets $\{a, b\}$ and $\{c, d\}$ are disjoint, no such path exists.}}{4}{}\protected@file@percent } \newlabel{fig:flip-proof-case-two}{{2}{4}} \gdef \@abspage@last{4} diff --git a/papers/colored_edge_flip_classes/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk index 525a4a1..adbad6c 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"] 1778738955 "paper.tex" "paper.pdf" "paper" 1778738956 +["pdflatex"] 1778742998 "paper.tex" "paper.pdf" "paper" 1778742998 "/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 "" @@ -27,6 +27,7 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/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/cmmi9.pfb" 1248133631 36094 798f80770b3b148ceedd006d487db67c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" @@ -127,8 +128,8 @@ "/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" 1778738956 1677 32d4f0477b551efb4eb21134ead928b4 "pdflatex" - "paper.tex" 1778738936 12685 245aac37998ca821ebd3d5d5a691ec64 "" + "paper.aux" 1778742998 1730 4d7f3892dba90583ca8d3c50faea2157 "pdflatex" + "paper.tex" 1778742990 14020 e71c46937129b64f0afa621f9c03d8d9 "" (generated) "paper.aux" "paper.log" diff --git a/papers/colored_edge_flip_classes/paper.fls b/papers/colored_edge_flip_classes/paper.fls index 100e47f..f166da2 100644 --- a/papers/colored_edge_flip_classes/paper.fls +++ b/papers/colored_edge_flip_classes/paper.fls @@ -445,6 +445,7 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.p INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb +INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi9.pfb INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb diff --git a/papers/colored_edge_flip_classes/paper.log b/papers/colored_edge_flip_classes/paper.log index 77bb45a..18be000 100644 --- a/papers/colored_edge_flip_classes/paper.log +++ b/papers/colored_edge_flip_classes/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 02:27 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 03:16 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -500,37 +500,33 @@ cmr/bx/n/10 . 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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 +obtained by contracting $uv$ and discarding parallel edges. Then +$G_0 / uv$ is a simple planar graph on $|V(G_0)| - 1 \geq 4$ vertices +but is not in general a triangulation; triangulate its planar +embedding (by adding chords inside any non-triangular face) to obtain +a maximal planar graph $T$ on the same vertex set, with $G_0 / uv$ as +a spanning subgraph and $|V(T)| < |V(G_0)|$. By the minimality of +$G_0$, $T$ admits a proper $4$-coloring, which restricts to 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