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\begin{document}

\title{Maximal Planar Graph Edge Flipping}

%    Remove any unused author tags.

%    author one information
\author{Eric Bauerfeld}
\address{}
\curraddr{}
\email{}
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\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.  In particular,
the conclusion of
Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric} is consistent
with the prevailing trend: as $n$ grows, almost every maximal planar
graph on $n$ vertices is already excluded from flip-symmetry on
purely structural grounds, and any putative counterexample to the
Four Color Theorem is forced into a vanishingly small slice of the
class.

\end{document}

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