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\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{Four Color Theorem, edge flip, maximal planar graph,
Kempe chain, minimum counterexample}

\date{}

\dedicatory{}

\begin{abstract}
Suppose the Four Color Theorem fails, and let $G_0$ be a maximal
planar graph of minimum order with $\chi(G_0) \geq 5$.  Using an
edge-deletion argument together with a Kempe-chain swap, we show
that every graph in the flip neighborhood $\mathcal{N}(G_0)$ --- the
set of maximal planar graphs obtainable from $G_0$ by a single
admissible edge flip --- is $4$-colorable.  We also introduce the colored edge flip
class $\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a
proper $4$-coloring $\varphi$ of $H$, and record that
$G_0 \notin \mathcal{C}(H, \varphi)$ for any
$H \in \mathcal{N}(G_0)$ and any such $\varphi$.
\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.

Our principal observation
(Theorem~\ref{thm:flip-neighborhood-4colorable}) is that every graph
in the \emph{flip neighborhood} of $G_0$ --- the set
$\mathcal{N}(G_0)$ of maximal planar graphs obtainable from $G_0$ by
a single admissible edge flip --- is $4$-colorable.  In other words,
$G_0$ sits at the boundary of $4$-colorability: a single flip in any
direction yields a $4$-colorable graph.

To track this rigidity at the level of individual $4$-colorings, we
introduce the \emph{colored edge flip class}
$\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a proper
$4$-coloring $\varphi$ of $H$: the set of maximal planar graphs
reachable from $H$ by sequences of admissible edge flips that each
preserve $\varphi$.  Theorem~\ref{thm:no-colored-class-contains-G}
records that $G_0 \notin \mathcal{C}(H, \varphi)$ for any
$H \in \mathcal{N}(G_0)$ and any proper $4$-coloring $\varphi$ of
$H$.

\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 neighborhoods and colored edge flip classes}

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 neighborhood]\label{def:flip-neighborhood}
Let $G$ be a maximal planar graph.  The \emph{flip neighborhood} of
$G$ is the set
\[
  \mathcal{N}(G) \;=\; \bigl\{\, G^{\mathrm{flip}(uv)} : uv \in E(G)
    \text{ and the flip at } uv \text{ is admissible} \,\bigr\}
\]
of maximal planar graphs obtainable from $G$ by a single admissible
edge flip.
\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 $\mathcal{C}(G, \varphi)$ of maximal planar
graphs reachable from $G$ by some (possibly empty) sequence of
admissible edge flips, each of which leaves $\varphi$ a proper
$4$-coloring of the resulting graph.  Explicitly,
$H \in \mathcal{C}(G, \varphi)$ iff there exist graphs
$G = G_0, G_1, \ldots, G_k = H$ such that for each $0 \leq i < k$,
$G_{i+1} = G_i^{\mathrm{flip}(u_i v_i)}$ for some
$u_i v_i \in E(G_i)$ whose flip is admissible in $G_i$ and whose
opposite vertices $w_i, x_i$ satisfy
$\varphi(w_i) \neq \varphi(x_i)$.
\end{definition}

\section{The flip neighborhood of a minimum-order 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.  Then
$G_0 / uv$ is a simple planar graph on $|V(G_0)| - 1 \geq 4$ vertices
but is not in general a triangulation; triangulate its planar
embedding (by adding chords inside any non-triangular face) to obtain
a maximal planar graph $T$ on the same vertex set, with $G_0 / uv$ as
a spanning subgraph and $|V(T)| < |V(G_0)|$.  By the minimality of
$G_0$, $T$ admits a proper $4$-coloring, which restricts to 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:flip-neighborhood-4colorable}
Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$.
Then every $H \in \mathcal{N}(G)$ 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:flip-neighborhood-4colorable}: $u, v$ 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}

\begin{theorem}\label{thm:no-colored-class-contains-G}
Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$.
Then for every $H \in \mathcal{N}(G)$ and every proper $4$-coloring
$\varphi$ of $H$,
\[
  G \;\notin\; \mathcal{C}(H, \varphi).
\]
\end{theorem}

\begin{proof}
Suppose, for contradiction, that $G \in \mathcal{C}(H, \varphi)$ for
some $H \in \mathcal{N}(G)$ and some proper $4$-coloring $\varphi$ of
$H$.  By Definition~\ref{def:colored-flip-class}, there exists a
sequence of maximal planar graphs $H = H_0, H_1, \ldots, H_k = G$ in
which each $H_{i+1}$ is obtained from $H_i$ by an admissible edge
flip that leaves $\varphi$ a proper $4$-coloring of $H_{i+1}$.  By
induction on $i$, $\varphi$ is a proper $4$-coloring of every $H_i$;
in particular, $\varphi$ is a proper $4$-coloring of $H_k = G$.  But
$\chi(G) \geq 5$ admits no such coloring, a contradiction.
\end{proof}

\end{document}

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