didericis
6c7bd9e0db
For G_0 a minimum-order 5-chromatic maximal planar graph and any 4-coloring of G_0 - uv, the endpoints u, v must share a color, and the color classes pairing that color with each of two other colors must each induce a u-v path. The Kempe-chain parts follow from a standard swap-on-component contradiction against the shared-color claim. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
358 lines
14 KiB
TeX
358 lines
14 KiB
TeX
%% filename: amsart-template.tex
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%% Copyright 2008-2010, 2014 American Mathematical Society.
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% AMS-LaTeX v.2 template for use with amsart
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% Remove any commented or uncommented macros you do not use.
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\documentclass{amsart}
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\begin{document}
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\title{Flip Symmetric Maximal Planar Graphs}
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% Remove any unused author tags.
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% author one information
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\author{Eric Bauerfeld}
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\address{}
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\curraddr{}
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\email{}
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\thanks{}
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\subjclass[2010]{Primary }
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\keywords{}
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\date{}
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\dedicatory{}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\section{Motivation}
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The Four Color Theorem asserts that every planar graph is properly
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$4$-colorable, or equivalently that no maximal planar graph $G$
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satisfies $\chi(G) \geq 5$. Suppose, towards a contradiction, that
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such a graph exists; let $G_0$ be one of minimum order. Any structural
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property shared by every maximal planar graph $H$ with $|V(H)| =
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|V(G_0)|$ is then automatically inherited by $G_0$, and any property
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\emph{not} satisfied by $G_0$ excludes a portion of the class of
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maximal planar graphs from playing the role of a minimum
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counterexample.
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This paper investigates one such property: invariance under an
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admissible edge flip. We call a maximal planar graph $G$
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\emph{flip-symmetric} when some admissible flip at an edge of $G$
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returns a graph isomorphic to $G$. Our principal observation
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(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that a
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minimum-order $5$-chromatic maximal planar graph cannot be
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flip-symmetric, so the search for a counterexample to the Four Color
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Theorem may, in principle, be confined to the complement of the class
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$\mathcal{F}$ of flip-symmetric graphs. This raises a quantitative
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question --- how large is $\mathcal{F}$? --- which we address
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empirically in Section~\ref{sec:frequency} by an exhaustive census of
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maximal planar graphs of small order.
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\section{Preliminaries}
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Let $G$ be a maximal planar graph with $|V(G)| \geq 4$, embedded in the
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plane so that every face --- including the outer face --- is a triangle.
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Every edge $uv \in E(G)$ is then shared by exactly two triangular faces
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$uvw$ and $uvx$ whose union is a quadrilateral $uwvx$ with diagonal $uv$.
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\begin{definition}[Edge flip]
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Let $G$ be a maximal planar graph and let $uv \in E(G)$ be an edge whose
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two incident triangular faces are $uvw$ and $uvx$. The \emph{edge flip}
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(or \emph{diagonal flip}) at $uv$ is the operation that deletes the edge
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$uv$ and inserts the edge $wx$ in its place, replacing the two triangles
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$uvw$ and $uvx$ by the two triangles $uwx$ and $vwx$. The flip is
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\emph{admissible} if $wx \notin E(G)$; otherwise the resulting multigraph
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is not simple and the flip is forbidden.
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\end{definition}
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\begin{figure}[h]
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\centering
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\begin{tikzpicture}[
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every node/.style={circle, fill=black, inner sep=1.5pt},
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label distance=2pt,
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scale=1.2
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]
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% --- before flip ---
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\begin{scope}[xshift=0cm]
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\node[label=left:$u$] (u) at (0,0) {};
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\node[label=right:$v$] (v) at (2,0) {};
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\node[label=above:$w$] (w) at (1,1) {};
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\node[label=below:$x$] (x) at (1,-1) {};
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\draw (u) -- (w) -- (v) -- (x) -- (u);
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\draw[very thick] (u) -- (v);
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\node[draw=none, fill=none] at (1,-1.6) {before};
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\end{scope}
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% --- arrow ---
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\draw[->, very thick, shorten >=2pt, shorten <=2pt]
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(2.6,0) -- node[draw=none, fill=none, above]{flip $uv$} (3.8,0);
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% --- after flip ---
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\begin{scope}[xshift=4.4cm]
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\node[label=left:$u$] (u2) at (0,0) {};
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\node[label=right:$v$] (v2) at (2,0) {};
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\node[label=above:$w$] (w2) at (1,1) {};
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\node[label=below:$x$] (x2) at (1,-1) {};
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\draw (u2) -- (w2) -- (v2) -- (x2) -- (u2);
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\draw[very thick] (w2) -- (x2);
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\node[draw=none, fill=none] at (1,-1.6) {after};
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\end{scope}
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\end{tikzpicture}
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\caption{An edge flip replaces the diagonal $uv$ of the quadrilateral
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$uwvx$ with the diagonal $wx$.}
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\end{figure}
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\section{Flip-symmetric maximal planar graphs}
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For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$
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with incident triangles $uvw$, $uvx$, write
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\[
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G^{\mathrm{flip}(uv)} \;=\; \bigl(V(G),\,(E(G)\setminus\{uv\})\cup\{wx\}\bigr)
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\]
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for the graph obtained from $G$ by flipping $uv$.
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\begin{definition}[Flip-symmetric graph]\label{def:flip-symmetric}
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A maximal planar graph $G$ is \emph{flip-symmetric} if there exists an
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admissible edge $uv \in E(G)$ such that
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$G^{\mathrm{flip}(uv)} \cong G$. We write $\mathcal{F}$ for the class
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of flip-symmetric maximal planar graphs.
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\end{definition}
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\section{A minimal four-colorable counterexample}
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\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric}
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Let $G$ be a maximal planar graph of minimum order among all maximal
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planar graphs $H$ with $\chi(H) \geq 5$. Then $G \notin \mathcal{F}$;
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that is, $G$ is not flip-symmetric.
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\end{theorem}
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\section{Flip symmetry frequency}\label{sec:frequency}
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To gauge how restrictive flip-symmetry is, we performed an exhaustive
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census of maximal planar graphs of small order. For each
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$n \in \{4, 5, \dots, 14\}$ we enumerated every isomorphism class of
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maximal planar graph on $n$ vertices using \texttt{plantri} (invoked
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through SageMath as \texttt{graphs.planar\_graphs} with
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\texttt{minimum\_connectivity}~$=3$ and
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\texttt{maximum\_face\_size}~$=3$), and for each such $G$ we tested
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every admissible edge $uv \in E(G)$ for the existence of an isomorphism
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$G \cong G^{\mathrm{flip}(uv)}$. Writing $T_n$ for the total number of
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maximal planar graphs on $n$ vertices and
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$F_n = |\mathcal{F} \cap \{G : |V(G)| = n\}|$ for the number of
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flip-symmetric ones, the results are tabulated below.
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\begin{center}
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\begin{tabular}{r r r l}
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\hline
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$n$ & $T_n$ & $F_n$ & $F_n / T_n$ \\
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\hline
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$4$ & $1$ & $0$ & $0.000000$ \\
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$5$ & $1$ & $1$ & $1.000000$ \\
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$6$ & $2$ & $1$ & $0.500000$ \\
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$7$ & $5$ & $1$ & $0.200000$ \\
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$8$ & $14$ & $5$ & $0.357143$ \\
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$9$ & $50$ & $17$ & $0.340000$ \\
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$10$ & $233$ & $48$ & $0.206009$ \\
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$11$ & $1{,}249$ & $164$ & $0.131305$ \\
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$12$ & $7{,}595$ & $552$ & $0.072679$ \\
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$13$ & $49{,}566$ & $1{,}828$ & $0.036880$ \\
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$14$ & $339{,}722$& $6{,}164$ & $0.018144$ \\
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\hline
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\end{tabular}
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\end{center}
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From $n = 10$ onward the ratio $F_n / T_n$ decreases by a factor
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approaching $1/2$ at each step, suggesting that the density of
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flip-symmetric graphs among maximal planar graphs of order $n$ decays
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to zero --- empirically at a roughly geometric rate. This tempers
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the utility of
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Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}: although it
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guarantees that a minimum-order counterexample to the Four Color
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Theorem lies in the complement of $\mathcal{F}$, that complement
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already comprises nearly the entire class of maximal planar graphs
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on $n$ vertices once $n$ is moderately large. The structural
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exclusion offered by flip-symmetry therefore prunes a vanishingly
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small portion of the search space, and this property is unlikely on
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its own to be a productive avenue for narrowing the search for a
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counterexample.
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A natural follow-up question is whether the picture improves when one
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restricts attention to maximal planar graphs of minimum degree at
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least~$5$, the class to which any minimum-order $5$-chromatic graph
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necessarily belongs (a vertex of degree at most~$4$ admits a standard
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Kempe reduction). Writing $T^{(5)}_n$ and $F^{(5)}_n$ for the
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analogous counts within this subclass, we ran the same census after
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adding \texttt{minimum\_degree}~$=5$ to the \texttt{plantri}
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invocation, obtaining the table below.
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\begin{center}
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\begin{tabular}{r r r l}
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\hline
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$n$ & $T^{(5)}_n$ & $F^{(5)}_n$ & $F^{(5)}_n / T^{(5)}_n$ \\
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\hline
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$12$ & $1$ & $0$ & $0.000000$ \\
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$13$ & $0$ & $0$ & --- \\
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$14$ & $1$ & $0$ & $0.000000$ \\
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$15$ & $1$ & $0$ & $0.000000$ \\
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$16$ & $3$ & $1$ & $0.333333$ \\
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$17$ & $4$ & $1$ & $0.250000$ \\
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$18$ & $12$ & $2$ & $0.166667$ \\
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$19$ & $23$ & $5$ & $0.217391$ \\
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$20$ & $73$ & $12$ & $0.164384$ \\
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$21$ & $192$ & $27$ & $0.140625$ \\
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$22$ & $651$ & $51$ & $0.078341$ \\
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$23$ & $2{,}070$ & $120$ & $0.057971$ \\
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$24$ & $7{,}290$ & $273$ & $0.037449$ \\
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$25$ & $25{,}381$ & $598$ & $0.023561$ \\
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$26$ & $91{,}441$ & $1{,}341$ & $0.014665$ \\
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\hline
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\end{tabular}
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\end{center}
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The first flip-symmetric example in this subclass appears at $n = 16$.
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Beyond that, the density $F^{(5)}_n / T^{(5)}_n$ again decays toward
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zero, though at a noticeably gentler rate: the step-to-step ratio
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settles around $0.63$ rather than the $\approx\!1/2$ observed in the
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unrestricted census. The restriction to minimum degree~$5$ therefore
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preserves flip-symmetry slightly longer relative to the size of the
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subclass, but does not alter the qualitative conclusion: even within
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the minimum-degree-$5$ class --- which already contains every
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candidate minimum-order $5$-chromatic graph --- flip-symmetric
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examples become a vanishing fraction.
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\section{Further necessary properties of a minimal counterexample}
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The frequency data of Section~\ref{sec:frequency} look unflattering
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only when flip-symmetry is weighed against the full class of maximal
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planar graphs. The class that actually matters --- minimum-order
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$5$-chromatic triangulations that also resist every Kempe-style
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reduction --- is far thinner, and flip-symmetry may exclude a
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substantially larger fraction of it if the configurations it removes
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overlap those responsible for Kempe reducibility. We therefore turn
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to identifying further necessary properties of a minimum-order
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$5$-chromatic maximal planar graph, of which flip-asymmetry is the
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first.
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\section{Edge-deletion subgraphs}
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\begin{definition}[Edge-deletion subgraph]\label{def:edge-deletion}
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Let $G$ be a maximal planar graph and $uv \in E(G)$. The
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\emph{edge-deletion subgraph at $uv$} is the spanning subgraph
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$G - uv = (V(G),\,E(G) \setminus \{uv\})$. Write
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$\mathcal{D}(G) = \{G - uv : uv \in E(G)\}$.
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\end{definition}
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\begin{theorem}\label{thm:edge-deletion-4colorable}
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Let $G_0$ be a maximal planar graph of minimum order with
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$\chi(G_0) \geq 5$. Then every $H \in \mathcal{D}(G_0)$ is
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$4$-colorable.
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\end{theorem}
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\begin{proof}
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Fix $uv \in E(G_0)$ and let $G_0 / uv$ denote the simple planar graph
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obtained by contracting $uv$ and discarding parallel edges. Since
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$|V(G_0/uv)| = |V(G_0)| - 1$, the minimality of $G_0$ supplies a
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proper $4$-coloring $c$ of $G_0 / uv$. Let $z$ be the contracted
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vertex and define $c'\colon V(G_0) \to \{1,2,3,4\}$ by
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$c'(u) = c'(v) = c(z)$ and $c'(y) = c(y)$ for $y \notin \{u, v\}$.
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Every edge of $G_0 - uv$ is either disjoint from $\{u, v\}$ or
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incident to exactly one of them; in either case the corresponding
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edge of $G_0 / uv$ has distinct endpoints under $c$, so $c'$ assigns
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its endpoints distinct colors. The edge $uv$ itself is absent from
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$G_0 - uv$, so $c'$ is a proper $4$-coloring of $G_0 - uv$.
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\end{proof}
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\begin{theorem}\label{thm:edge-deletion-coloring-structure}
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Let $G_0$ be a maximal planar graph of minimum order with
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$\chi(G_0) \geq 5$, fix $uv \in E(G_0)$, and let $\varphi$ be any
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proper $4$-coloring of $G_0 - uv$. Write $a = \varphi(u)$ and let
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$b, c, d$ denote the three remaining colors. Then:
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\begin{enumerate}
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\item $\varphi(v) = a$;
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\item the subgraph of $G_0 - uv$ induced by the vertices of color
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$a$ or $b$ contains a path from $u$ to $v$;
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\item the subgraph of $G_0 - uv$ induced by the vertices of color
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$a$ or $c$ contains a path from $u$ to $v$.
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\end{enumerate}
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\end{theorem}
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\begin{proof}
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(1) If $\varphi(v) \neq a$ then $\varphi$ is already a proper
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$4$-coloring of $G_0$, since the only edge of $G_0$ absent from
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$G_0 - uv$ is $uv$ and its endpoints have distinct colors. This
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contradicts $\chi(G_0) \geq 5$, so $\varphi(v) = a$.
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(2) Suppose, for contradiction, that $u$ and $v$ lie in distinct
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connected components of the subgraph of $G_0 - uv$ induced by the
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color classes $a$ and $b$. Let $C$ be the component containing $u$,
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and define $\varphi'\colon V(G_0) \to \{a,b,c,d\}$ by swapping colors
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$a \leftrightarrow b$ on $C$ and leaving every other vertex
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unchanged. Then $\varphi'$ is a proper $4$-coloring of $G_0 - uv$
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with $\varphi'(u) = b$ and $\varphi'(v) = a$, contradicting part~(1)
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applied to $\varphi'$.
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(3) Identical to (2) with $c$ in place of $b$.
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\end{proof}
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\end{document}
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