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\begin{document}

\title{Flip Symmetric Maximal Planar Graphs}

%    Remove any unused author tags.

%    author one information
\author{Eric Bauerfeld}
\address{}
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\email{}
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\subjclass[2010]{Primary }

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\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.

\section{Further necessary properties of a minimal counterexample}

The frequency data of Section~\ref{sec:frequency} look unflattering
only when flip-symmetry is weighed against the full class of maximal
planar graphs.  The class that actually matters --- minimum-order
$5$-chromatic triangulations that also resist every Kempe-style
reduction --- is far thinner, and flip-symmetry may exclude a
substantially larger fraction of it if the configurations it removes
overlap those responsible for Kempe reducibility.  We therefore turn
to identifying further necessary properties of a minimum-order
$5$-chromatic maximal planar graph, of which flip-asymmetry is the
first.

\section{Edge-deletion subgraphs}

\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{theorem}\label{thm: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{theorem}

\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{theorem}\label{thm: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{theorem}

\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}

\end{document}

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