From 53a9192f65e955f5df9c3a9dc5359cfcc31d27a6 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 03:22:33 -0400 Subject: [PATCH] Drop flip-symmetry framing Remove the flip-symmetric definition, the class $\mathcal{F}$, and all references to flip-symmetry from the abstract, motivation, and section 3 title. Section 3 is renamed to reflect what remains: the flip neighborhood and the colored edge flip class. The principal theorem's label is renamed to thm:flip-neighborhood-4colorable to match its statement. Co-Authored-By: Claude Opus 4.7 --- papers/colored_edge_flip_classes/paper.aux | 15 +++++---- .../paper.fdb_latexmk | 6 ++-- papers/colored_edge_flip_classes/paper.log | 18 ++++------- papers/colored_edge_flip_classes/paper.pdf | Bin 247502 -> 246274 bytes papers/colored_edge_flip_classes/paper.tex | 29 +++++------------- 5 files changed, 23 insertions(+), 45 deletions(-) diff --git a/papers/colored_edge_flip_classes/paper.aux b/papers/colored_edge_flip_classes/paper.aux index 0524451..31d94f6 100644 --- a/papers/colored_edge_flip_classes/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -2,15 +2,14 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Preliminaries}}{1}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces An edge flip replaces the diagonal $uv$ of the quadrilateral $uwvx$ with the diagonal $wx$.}}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip-symmetric maximal planar graphs}}{2}{}\protected@file@percent } -\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}}{3}{}\protected@file@percent } -\newlabel{def:edge-deletion}{{4.1}{3}} -\newlabel{lem:edge-deletion-4colorable}{{4.2}{3}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip neighborhoods and colored edge flip classes}}{2}{}\protected@file@percent } +\newlabel{def:flip-neighborhood}{{3.1}{2}} +\newlabel{def:colored-flip-class}{{3.2}{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}} \newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}} -\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}} +\newlabel{thm:flip-neighborhood-4colorable}{{4.4}{3}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} diff --git a/papers/colored_edge_flip_classes/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk index 408afa8..ad32bcd 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"] 1778743194 "paper.tex" "paper.pdf" "paper" 1778743195 +["pdflatex"] 1778743331 "paper.tex" "paper.pdf" "paper" 1778743331 "/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 "" @@ -131,8 +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 "" - "paper.aux" 1778743195 1746 ea19789a676d7a11f10d4bf7271801ee "pdflatex" - "paper.tex" 1778743189 15567 99b4ce65094b56da9e5ecd2e093a7331 "" + "paper.aux" 1778743331 1709 057e58fcb5472314b0a7029f2c0f7505 "pdflatex" + "paper.tex" 1778743323 14730 0431b5dd1f68c135b8365d9286869b8f "" (generated) "paper.aux" "paper.log" diff --git a/papers/colored_edge_flip_classes/paper.log b/papers/colored_edge_flip_classes/paper.log index 648ee2f..ede1621 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 03:19 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 03:22 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -486,18 +486,12 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -Overfull \hbox (6.71799pt too wide) in paragraph at lines 195--199 -[]\OT1/cmr/bx/n/10 Definition 3.1 \OT1/cmr/m/n/10 (Flip-sym-met-ric graph)\OT1/ -cmr/bx/n/10 . 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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 +admissible edge flip --- is $4$-colorable. 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 @@ -104,19 +103,12 @@ maximal planar graphs from playing the role of a minimum counterexample. Our principal observation -(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that -every graph in the \emph{flip neighborhood} of $G_0$ --- the set +(Theorem~\ref{thm:flip-neighborhood-4colorable}) is that every graph +in the \emph{flip neighborhood} of $G_0$ --- the set $\mathcal{N}(G_0)$ of maximal planar graphs obtainable from $G_0$ by a single admissible edge flip --- is $4$-colorable. In other words, $G_0$ sits at the boundary of $4$-colorability: a single flip in any -direction yields a $4$-colorable graph. As an immediate corollary, -no such $G_0$ is \emph{flip-symmetric}, where we call a maximal -planar graph $G$ flip-symmetric when some admissible flip at an edge -of $G$ returns a graph isomorphic to $G$; if any flip of $G_0$ were -to return $G_0$, that flip would witness $G_0$ as $4$-colorable. The -search for a counterexample to the Four Color Theorem may therefore -be confined to the complement of the class $\mathcal{F}$ of -flip-symmetric maximal planar graphs. +direction yields a $4$-colorable graph. To track this rigidity at the level of individual $4$-colorings, we introduce the \emph{colored edge flip class} @@ -182,7 +174,7 @@ is not simple and the flip is forbidden. $uwvx$ with the diagonal $wx$.} \end{figure} -\section{Flip-symmetric maximal planar graphs} +\section{Flip neighborhoods and colored edge flip classes} For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$ with incident triangles $uvw$, $uvx$, write @@ -191,13 +183,6 @@ with incident triangles $uvw$, $uvx$, write \] for the graph obtained from $G$ by flipping $uv$. -\begin{definition}[Flip-symmetric graph]\label{def:flip-symmetric} -A maximal planar graph $G$ is \emph{flip-symmetric} if there exists an -admissible edge $uv \in E(G)$ such that -$G^{\mathrm{flip}(uv)} \cong G$. We write $\mathcal{F}$ for the class -of flip-symmetric maximal planar graphs. -\end{definition} - \begin{definition}[Flip neighborhood]\label{def:flip-neighborhood} Let $G$ be a maximal planar graph. The \emph{flip neighborhood} of $G$ is the set @@ -290,7 +275,7 @@ applied to $\varphi'$. (3) Identical to (2) with $c$ in place of $b$. \end{proof} -\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric} +\begin{theorem}\label{thm:flip-neighborhood-4colorable} Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$. Then every $H \in \mathcal{N}(G)$ is $4$-colorable. \end{theorem} @@ -356,7 +341,7 @@ reducing to Case~1. \node[font=\small] at (3.5, 2.6) {$\{a, b\}$-Kempe path $P$}; \end{tikzpicture} \caption{Case~2 of the proof of -Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}: $u, v$ share +Theorem~\ref{thm:flip-neighborhood-4colorable}: $u, v$ 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