Add edge-deletion subgraph 4-colorability for a minimal counterexample

Defines D(G) as the family of single-edge-deletion spanning subgraphs
of a maximal planar graph G, and shows that when G_0 is a minimum-order
5-chromatic maximal planar graph every member of D(G_0) is 4-colorable,
via a coloring pulled back from the smaller minor G_0/uv.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

papers/flip_symmetric_maximal_planar_graphs/paper.tex

@@ -289,6 +289,35 @@ 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}
+
 \end{document}
 
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