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 <noreply@anthropic.com>
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\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}}
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\@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 }
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\newlabel{fig:flip-proof-case-two}{{2}{4}}
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%% 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
|
||||
%-----------------------------------------------------------------------
|
||||
Binary file not shown.
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|
||||
%% 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{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
|
||||
%-----------------------------------------------------------------------
|
||||
Reference in New Issue
Block a user