From c59d2e95e1c0cb32cdb119791913187867b1cbca Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 02:03:17 -0400 Subject: [PATCH] Refocus paper on colored edge flip classes; drop frequency census Renames the paper (and its directory) to reflect the shift in emphasis toward the colored edge flip class introduced last commit, and removes the flip-symmetry frequency section: the unsigned-flip census was a digression that the new framing no longer needs, and its prose conclusion was at odds with the direction the paper is heading. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 12 +- .../paper.fdb_latexmk | 17 +- .../paper.fls | 15 +- .../paper.log | 61 ++-- papers/colored_edge_flip_classes/paper.pdf | Bin 0 -> 199834 bytes papers/colored_edge_flip_classes/paper.tex | 331 ++++++++++++++++++ .../paper.pdf | Bin 164862 -> 0 bytes .../paper.tex | 283 --------------- 8 files changed, 399 insertions(+), 320 deletions(-) rename papers/{flip_symmetric_maximal_planar_graphs => colored_edge_flip_classes}/paper.aux (52%) rename papers/{flip_symmetric_maximal_planar_graphs => colored_edge_flip_classes}/paper.fdb_latexmk (92%) rename papers/{flip_symmetric_maximal_planar_graphs => colored_edge_flip_classes}/paper.fls (97%) rename papers/{flip_symmetric_maximal_planar_graphs => colored_edge_flip_classes}/paper.log (90%) create mode 100644 papers/colored_edge_flip_classes/paper.pdf create mode 100644 papers/colored_edge_flip_classes/paper.tex delete mode 100644 papers/flip_symmetric_maximal_planar_graphs/paper.pdf delete mode 100644 papers/flip_symmetric_maximal_planar_graphs/paper.tex diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/colored_edge_flip_classes/paper.aux similarity index 52% rename from papers/flip_symmetric_maximal_planar_graphs/paper.aux rename to papers/colored_edge_flip_classes/paper.aux index b17b3d3..86df3ab 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -4,13 +4,17 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip-symmetric maximal planar graphs}}{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 } \newlabel{def:flip-symmetric}{{3.1}{2}} +\newlabel{def:colored-flip-class}{{3.2}{2}} \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{A minimal four-colorable counterexample}}{2}{}\protected@file@percent } -\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}} +\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{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\gdef \@abspage@last{3} +\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{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/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk similarity index 92% rename from papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk rename to papers/colored_edge_flip_classes/paper.fdb_latexmk index 8febd69..3b35971 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk +++ b/papers/colored_edge_flip_classes/paper.fdb_latexmk @@ -1,8 +1,9 @@ # Fdb version 3 -["pdflatex"] 1778734485 "paper.tex" "paper.pdf" "paper" 1778734486 +["pdflatex"] 1778738215 "paper.tex" "paper.pdf" "paper" 1778738215 "/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 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex9.tfm" 1246382020 996 a18840b13b499c08ac2de96a99eda4bc "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a "" @@ -13,25 +14,31 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi9.tfm" 1136768653 1524 d89e2d087a9828407a196f428428ef4a "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr9.tfm" 1136768653 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy9.tfm" 1136768653 1116 25a7bf822c58caf309a702ef79f4afbb "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" - 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a/papers/colored_edge_flip_classes/paper.tex b/papers/colored_edge_flip_classes/paper.tex new file mode 100644 index 0000000..9baf0ab --- /dev/null +++ b/papers/colored_edge_flip_classes/paper.tex @@ -0,0 +1,331 @@ +%% filename: amsart-template.tex +%% version: 1.1 +%% date: 2014/07/24 +%% +%% American Mathematical Society +%% Technical Support +%% Publications Technical Group +%% 201 Charles Street +%% Providence, RI 02904 +%% USA +%% tel: (401) 455-4080 +%% (800) 321-4267 (USA and Canada only) +%% fax: (401) 331-3842 +%% email: tech-support@ams.org +%% +%% Copyright 2008-2010, 2014 American Mathematical Society. +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3c +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3c or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is the American Mathematical +%% Society. +%% +%% ==================================================================== + +% AMS-LaTeX v.2 template for use with amsart +% +% Remove any commented or uncommented macros you do not use. + +\documentclass{amsart} + +\usepackage{tikz} + +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{conjecture}[theorem]{Conjecture} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{xca}[theorem]{Exercise} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} + +\numberwithin{equation}{section} + +\begin{document} + +\title{Colored Edge Flip Classes} + +% Remove any unused author tags. + +% author one information +\author{Eric Bauerfeld} +\address{} +\curraddr{} +\email{} +\thanks{} + + +\subjclass[2010]{Primary } + +\keywords{} + +\date{} + +\dedicatory{} + +\begin{abstract} +\end{abstract} + +\maketitle + +\section{Motivation} + +The Four Color Theorem asserts that every planar graph is properly +$4$-colorable, or equivalently that no maximal planar graph $G$ +satisfies $\chi(G) \geq 5$. Suppose, towards a contradiction, that +such a graph exists; let $G_0$ be one of minimum order. Any structural +property shared by every maximal planar graph $H$ with $|V(H)| = +|V(G_0)|$ is then automatically inherited by $G_0$, and any property +\emph{not} satisfied by $G_0$ excludes a portion of the class of +maximal planar graphs from playing the role of a minimum +counterexample. + +This paper investigates one such property: behavior under an edge +flip. Our principal observation +(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that +every edge flip of a minimum-order $5$-chromatic maximal planar graph +yields a $4$-colorable graph. In particular, no such graph 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 $G$. 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 graphs. + +\section{Preliminaries} + +Let $G$ be a maximal planar graph with $|V(G)| \geq 4$, embedded in the +plane so that every face --- including the outer face --- is a triangle. +Every edge $uv \in E(G)$ is then shared by exactly two triangular faces +$uvw$ and $uvx$ whose union is a quadrilateral $uwvx$ with diagonal $uv$. + +\begin{definition}[Edge flip] +Let $G$ be a maximal planar graph and let $uv \in E(G)$ be an edge whose +two incident triangular faces are $uvw$ and $uvx$. The \emph{edge flip} +(or \emph{diagonal flip}) at $uv$ is the operation that deletes the edge +$uv$ and inserts the edge $wx$ in its place, replacing the two triangles +$uvw$ and $uvx$ by the two triangles $uwx$ and $vwx$. The flip is +\emph{admissible} if $wx \notin E(G)$; otherwise the resulting multigraph +is not simple and the flip is forbidden. +\end{definition} + +\begin{figure}[h] +\centering +\begin{tikzpicture}[ + every node/.style={circle, fill=black, inner sep=1.5pt}, + label distance=2pt, + scale=1.2 +] + % --- before flip --- + \begin{scope}[xshift=0cm] + \node[label=left:$u$] (u) at (0,0) {}; + \node[label=right:$v$] (v) at (2,0) {}; + \node[label=above:$w$] (w) at (1,1) {}; + \node[label=below:$x$] (x) at (1,-1) {}; + \draw (u) -- (w) -- (v) -- (x) -- (u); + \draw[very thick] (u) -- (v); + \node[draw=none, fill=none] at (1,-1.6) {before}; + \end{scope} + + % --- arrow --- + \draw[->, very thick, shorten >=2pt, shorten <=2pt] + (2.6,0) -- node[draw=none, fill=none, above]{flip $uv$} (3.8,0); + + % --- after flip --- + \begin{scope}[xshift=4.4cm] + \node[label=left:$u$] (u2) at (0,0) {}; + \node[label=right:$v$] (v2) at (2,0) {}; + \node[label=above:$w$] (w2) at (1,1) {}; + \node[label=below:$x$] (x2) at (1,-1) {}; + \draw (u2) -- (w2) -- (v2) -- (x2) -- (u2); + \draw[very thick] (w2) -- (x2); + \node[draw=none, fill=none] at (1,-1.6) {after}; + \end{scope} +\end{tikzpicture} +\caption{An edge flip replaces the diagonal $uv$ of the quadrilateral +$uwvx$ with the diagonal $wx$.} +\end{figure} + +\section{Flip-symmetric maximal planar graphs} + +For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$ +with incident triangles $uvw$, $uvx$, write +\[ + G^{\mathrm{flip}(uv)} \;=\; \bigl(V(G),\,(E(G)\setminus\{uv\})\cup\{wx\}\bigr) +\] +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}[Colored edge flip class]\label{def:colored-flip-class} +Let $G$ be a maximal planar graph and let $\varphi$ be a proper +$4$-coloring of $G$. The \emph{colored edge flip class} of +$(G, \varphi)$ is the set +\[ + \mathcal{C}(G, \varphi) \;=\; \bigl\{\, G^{\mathrm{flip}(uv)} : + uv \in E(G),\ \text{the flip at } uv \text{ is admissible, and}\ + \varphi(w) \neq \varphi(x) \,\bigr\}, +\] +where $w, x$ are the third vertices of the two triangular faces of +$G$ containing $uv$. Equivalently, $\mathcal{C}(G, \varphi)$ is the +set of graphs obtained from $G$ by an admissible edge flip under +which $\varphi$ remains a proper $4$-coloring. +\end{definition} + +\section{A minimal four-colorable counterexample} + +\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{lemma}\label{lem: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{lemma} + +\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} + +\begin{lemma}\label{lem: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{lemma} + +\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} + +\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric} +Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$. +Then for every edge $e \in E(G)$, the graph induced by an edge flip +of $e$ is $4$-colorable. +\end{theorem} + +\begin{proof} +Fix an edge $e = uv \in E(G)$, and let $F_0, F_1$ be the two +triangular faces of $G$ incident to $e$, so that +$\{w, x\} = \bigl(V(F_0) \cup V(F_1)\bigr) \setminus \{u, v\}$. By +Lemma~\ref{lem:edge-deletion-4colorable}, $G - e$ admits a proper +$4$-coloring $\varphi$. + +\smallskip +\noindent\emph{Case 1: $\varphi(w) \neq \varphi(x)$.} Then $\varphi$ +is also a proper $4$-coloring of the graph induced by the edge flip +of $e$. + +\smallskip +\noindent\emph{Case 2: $\varphi(w) = \varphi(x)$.} Set +$a = \varphi(u)$; by Lemma~\ref{lem:edge-deletion-coloring-structure}(1), +$\varphi(v) = a$ as well, and the edges $uw, vw \in E(G - e)$ force +$\varphi(w) \neq a$. Choose a color $b \notin \{a, \varphi(w)\}$. +By Lemma~\ref{lem:edge-deletion-coloring-structure}, there is a path +$P$ from $u$ to $v$ in the subgraph of $G - e$ induced by the +vertices of color $a$ or $b$. Let +$\{c, d\} = \{1, 2, 3, 4\} \setminus \{a, b\}$; then +$\varphi(w) = \varphi(x) \in \{c, d\}$. + +Any path from $w$ to $x$ in the subgraph of $G - e$ induced by the +vertices of color $c$ or $d$ would, in the plane embedding of +$G - e$, cross $P$; but its vertices have colors in +$\{c, d\}$ and the vertices of $P$ have colors in $\{a, b\}$, and +these sets are disjoint, so the two paths share no vertex. Hence +$w$ and $x$ lie in distinct connected components of the +$\{c, d\}$-colored subgraph of $G - e$. Swapping colors +$c \leftrightarrow d$ on the component containing $w$ yields a proper +$4$-coloring of $G - e$ in which $\varphi(w) \neq \varphi(x)$, +reducing to Case~1. +\end{proof} + +\begin{figure}[h] +\centering +\begin{tikzpicture}[ + vertex/.style={circle, draw, minimum size=18pt, inner sep=0pt, font=\small}, + scale=1.0 +] + % --- u and v with color a --- + \node[vertex, fill=red!25, label=left:$u$] (u) at (0, 0) {$a$}; + \node[vertex, fill=red!25, label=right:$v$] (v) at (7, 0) {$a$}; + + % --- w (above) and x (below): both colored c in Case 2 --- + \node[vertex, fill=green!30, label=above:$w$] (w) at (3.5, 0.6) {$c$}; + \node[vertex, fill=green!30, label=below:$x$] (x) at (3.5, -0.6) {$c$}; + \draw (u) -- (w) -- (v); + \draw (u) -- (x) -- (v); + + % --- {a, b}-Kempe path P from u to v --- + \node[vertex, fill=blue!25] (b1) at (1.5, 1.7) {$b$}; + \node[vertex, fill=red!25] (a1) at (3.5, 2.0) {$a$}; + \node[vertex, fill=blue!25] (b2) at (5.5, 1.7) {$b$}; + \draw (u) -- (b1) -- (a1) -- (b2) -- (v); + + % Path label + \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 +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.} +\label{fig:flip-proof-case-two} +\end{figure} + +\end{document} + +%----------------------------------------------------------------------- +% End of amsart-template.tex +%----------------------------------------------------------------------- diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.pdf b/papers/flip_symmetric_maximal_planar_graphs/paper.pdf deleted file mode 100644 index 072cd63327ca9f3ad72f587944a86c0f40238cf6..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 164862 zcma&NLzFH|u&!CQ?fS~LZCkr++qP}nw(VWE?W$e2-GBG$Gdg#0a}=3_h_y0u@WguG zBvTX>r(>dLgCUz=9$ANB2QUKcjjUjJd0`l2P3_EGEC9@0>`Z|FZ^1B#TiUpoIsq8O zZ46yZMNN(EO-y0<`C*)0oJAZu%NX&P!{NzG9U{C&(h(a1KQyqqmSpOWAHIdDE^L#EE&n4MppG(SzhhF_ZR 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Project Public License, either version 1.3c -%% of this license or (at your option) any later version. -%% The latest version of this license is in -%% http://www.latex-project.org/lppl.txt -%% and version 1.3c or later is part of all distributions of LaTeX -%% version 2005/12/01 or later. -%% -%% This work has the LPPL maintenance status `maintained'. -%% -%% The Current Maintainer of this work is the American Mathematical -%% Society. -%% -%% ==================================================================== - -% AMS-LaTeX v.2 template for use with amsart -% -% Remove any commented or uncommented macros you do not use. - -\documentclass{amsart} - -\usepackage{tikz} - -\newtheorem{theorem}{Theorem}[section] -\newtheorem{lemma}[theorem]{Lemma} -\newtheorem{conjecture}[theorem]{Conjecture} - -\theoremstyle{definition} -\newtheorem{definition}[theorem]{Definition} -\newtheorem{example}[theorem]{Example} -\newtheorem{xca}[theorem]{Exercise} - -\theoremstyle{remark} -\newtheorem{remark}[theorem]{Remark} - -\numberwithin{equation}{section} - -\begin{document} - -\title{Flip Symmetric Maximal Planar Graphs} - -% Remove any unused author tags. - -% author one information -\author{Eric Bauerfeld} -\address{} -\curraddr{} -\email{} -\thanks{} - - -\subjclass[2010]{Primary } - -\keywords{} - -\date{} - -\dedicatory{} - -\begin{abstract} -\end{abstract} - -\maketitle - -\section{Motivation} - -The Four Color Theorem asserts that every planar graph is properly -$4$-colorable, or equivalently that no maximal planar graph $G$ -satisfies $\chi(G) \geq 5$. Suppose, towards a contradiction, that -such a graph exists; let $G_0$ be one of minimum order. Any structural -property shared by every maximal planar graph $H$ with $|V(H)| = -|V(G_0)|$ is then automatically inherited by $G_0$, and any property -\emph{not} satisfied by $G_0$ excludes a portion of the class of -maximal planar graphs from playing the role of a minimum -counterexample. - -This paper investigates one such property: invariance under an -admissible edge flip. We call a maximal planar graph $G$ -\emph{flip-symmetric} when some admissible flip at an edge of $G$ -returns a graph isomorphic to $G$. Our principal observation -(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that a -minimum-order $5$-chromatic maximal planar graph cannot be -flip-symmetric, so the search for a counterexample to the Four Color -Theorem may, in principle, be confined to the complement of the class -$\mathcal{F}$ of flip-symmetric graphs. This raises a quantitative -question --- how large is $\mathcal{F}$? --- which we address -empirically in Section~\ref{sec:frequency} by an exhaustive census of -maximal planar graphs of small order. - -\section{Preliminaries} - -Let $G$ be a maximal planar graph with $|V(G)| \geq 4$, embedded in the -plane so that every face --- including the outer face --- is a triangle. -Every edge $uv \in E(G)$ is then shared by exactly two triangular faces -$uvw$ and $uvx$ whose union is a quadrilateral $uwvx$ with diagonal $uv$. - -\begin{definition}[Edge flip] -Let $G$ be a maximal planar graph and let $uv \in E(G)$ be an edge whose -two incident triangular faces are $uvw$ and $uvx$. The \emph{edge flip} -(or \emph{diagonal flip}) at $uv$ is the operation that deletes the edge -$uv$ and inserts the edge $wx$ in its place, replacing the two triangles -$uvw$ and $uvx$ by the two triangles $uwx$ and $vwx$. The flip is -\emph{admissible} if $wx \notin E(G)$; otherwise the resulting multigraph -is not simple and the flip is forbidden. -\end{definition} - -\begin{figure}[h] -\centering -\begin{tikzpicture}[ - every node/.style={circle, fill=black, inner sep=1.5pt}, - label distance=2pt, - scale=1.2 -] - % --- before flip --- - \begin{scope}[xshift=0cm] - \node[label=left:$u$] (u) at (0,0) {}; - \node[label=right:$v$] (v) at (2,0) {}; - \node[label=above:$w$] (w) at (1,1) {}; - \node[label=below:$x$] (x) at (1,-1) {}; - \draw (u) -- (w) -- (v) -- (x) -- (u); - \draw[very thick] (u) -- (v); - \node[draw=none, fill=none] at (1,-1.6) {before}; - \end{scope} - - % --- arrow --- - \draw[->, very thick, shorten >=2pt, shorten <=2pt] - (2.6,0) -- node[draw=none, fill=none, above]{flip $uv$} (3.8,0); - - % --- after flip --- - \begin{scope}[xshift=4.4cm] - \node[label=left:$u$] (u2) at (0,0) {}; - \node[label=right:$v$] (v2) at (2,0) {}; - \node[label=above:$w$] (w2) at (1,1) {}; - \node[label=below:$x$] (x2) at (1,-1) {}; - \draw (u2) -- (w2) -- (v2) -- (x2) -- (u2); - \draw[very thick] (w2) -- (x2); - \node[draw=none, fill=none] at (1,-1.6) {after}; - \end{scope} -\end{tikzpicture} -\caption{An edge flip replaces the diagonal $uv$ of the quadrilateral -$uwvx$ with the diagonal $wx$.} -\end{figure} - -\section{Flip-symmetric maximal planar graphs} - -For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$ -with incident triangles $uvw$, $uvx$, write -\[ - G^{\mathrm{flip}(uv)} \;=\; \bigl(V(G),\,(E(G)\setminus\{uv\})\cup\{wx\}\bigr) -\] -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} - -\section{A minimal four-colorable counterexample} - -\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric} -Let $G$ be a maximal planar graph of minimum order among all maximal -planar graphs $H$ with $\chi(H) \geq 5$. Then $G \notin \mathcal{F}$; -that is, $G$ is not flip-symmetric. -\end{theorem} - -\section{Flip symmetry frequency}\label{sec:frequency} - -To gauge how restrictive flip-symmetry is, we performed an exhaustive -census of maximal planar graphs of small order. For each -$n \in \{4, 5, \dots, 14\}$ we enumerated every isomorphism class of -maximal planar graph on $n$ vertices using \texttt{plantri} (invoked -through SageMath as \texttt{graphs.planar\_graphs} with -\texttt{minimum\_connectivity}~$=3$ and -\texttt{maximum\_face\_size}~$=3$), and for each such $G$ we tested -every admissible edge $uv \in E(G)$ for the existence of an isomorphism -$G \cong G^{\mathrm{flip}(uv)}$. Writing $T_n$ for the total number of -maximal planar graphs on $n$ vertices and -$F_n = |\mathcal{F} \cap \{G : |V(G)| = n\}|$ for the number of -flip-symmetric ones, the results are tabulated below. - -\begin{center} -\begin{tabular}{r r r l} -\hline -$n$ & $T_n$ & $F_n$ & $F_n / T_n$ \\ -\hline -$4$ & $1$ & $0$ & $0.000000$ \\ -$5$ & $1$ & $1$ & $1.000000$ \\ -$6$ & $2$ & $1$ & $0.500000$ \\ -$7$ & $5$ & $1$ & $0.200000$ \\ -$8$ & $14$ & $5$ & $0.357143$ \\ -$9$ & $50$ & $17$ & $0.340000$ \\ -$10$ & $233$ & $48$ & $0.206009$ \\ -$11$ & $1{,}249$ & $164$ & $0.131305$ \\ -$12$ & $7{,}595$ & $552$ & $0.072679$ \\ -$13$ & $49{,}566$ & $1{,}828$ & $0.036880$ \\ -$14$ & $339{,}722$& $6{,}164$ & $0.018144$ \\ -\hline -\end{tabular} -\end{center} - -From $n = 10$ onward the ratio $F_n / T_n$ decreases by a factor -approaching $1/2$ at each step, suggesting that the density of -flip-symmetric graphs among maximal planar graphs of order $n$ decays -to zero --- empirically at a roughly geometric rate. This tempers -the utility of -Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}: although it -guarantees that a minimum-order counterexample to the Four Color -Theorem lies in the complement of $\mathcal{F}$, that complement -already comprises nearly the entire class of maximal planar graphs -on $n$ vertices once $n$ is moderately large. The structural -exclusion offered by flip-symmetry therefore prunes a vanishingly -small portion of the search space, and this property is unlikely on -its own to be a productive avenue for narrowing the search for a -counterexample. - -A natural follow-up question is whether the picture improves when one -restricts attention to maximal planar graphs of minimum degree at -least~$5$, the class to which any minimum-order $5$-chromatic graph -necessarily belongs (a vertex of degree at most~$4$ admits a standard -Kempe reduction). Writing $T^{(5)}_n$ and $F^{(5)}_n$ for the -analogous counts within this subclass, we ran the same census after -adding \texttt{minimum\_degree}~$=5$ to the \texttt{plantri} -invocation, obtaining the table below. - -\begin{center} -\begin{tabular}{r r r l} -\hline -$n$ & $T^{(5)}_n$ & $F^{(5)}_n$ & $F^{(5)}_n / T^{(5)}_n$ \\ -\hline -$12$ & $1$ & $0$ & $0.000000$ \\ -$13$ & $0$ & $0$ & --- \\ -$14$ & $1$ & $0$ & $0.000000$ \\ -$15$ & $1$ & $0$ & $0.000000$ \\ -$16$ & $3$ & $1$ & $0.333333$ \\ -$17$ & $4$ & $1$ & $0.250000$ \\ -$18$ & $12$ & $2$ & $0.166667$ \\ -$19$ & $23$ & $5$ & $0.217391$ \\ -$20$ & $73$ & $12$ & $0.164384$ \\ -$21$ & $192$ & $27$ & $0.140625$ \\ -$22$ & $651$ & $51$ & $0.078341$ \\ -$23$ & $2{,}070$ & $120$ & $0.057971$ \\ -$24$ & $7{,}290$ & $273$ & $0.037449$ \\ -$25$ & $25{,}381$ & $598$ & $0.023561$ \\ -$26$ & $91{,}441$ & $1{,}341$ & $0.014665$ \\ -\hline -\end{tabular} -\end{center} - -The first flip-symmetric example in this subclass appears at $n = 16$. -Beyond that, the density $F^{(5)}_n / T^{(5)}_n$ again decays toward -zero, though at a noticeably gentler rate: the step-to-step ratio -settles around $0.63$ rather than the $\approx\!1/2$ observed in the -unrestricted census. The restriction to minimum degree~$5$ therefore -preserves flip-symmetry slightly longer relative to the size of the -subclass, but does not alter the qualitative conclusion: even within -the minimum-degree-$5$ class --- which already contains every -candidate minimum-order $5$-chromatic graph --- flip-symmetric -examples become a vanishing fraction. - -\end{document} - -%----------------------------------------------------------------------- -% End of amsart-template.tex -%-----------------------------------------------------------------------