Add Kempe-chain structure of 4-colorings of edge-deletion subgraphs

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>

papers/flip_symmetric_maximal_planar_graphs/paper.tex

@@ -318,6 +318,38 @@ 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}
 
 %-----------------------------------------------------------------------