Drop edge-deletion sections from flip-symmetry paper

Removes the "Further necessary properties of a minimal counterexample"
framing section and the "Edge-deletion subgraphs" section (definition,
4-colorability theorem, Kempe-chain structure theorem). The intended
empirical follow-up on this material did not produce a useful
discriminator, so the development is being shelved.

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

papers/flip_symmetric_maximal_planar_graphs/paper.aux

@@ -8,14 +8,9 @@
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 \newlabel{sec:frequency}{{5}{2}}
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 \newlabel{tocindent-1}{0pt}
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 \newlabel{tocindent3}{0pt}
-\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Edge-deletion subgraphs}}{4}{}\protected@file@percent }
+\gdef \@abspage@last{3}
-\newlabel{def:edge-deletion}{{7.1}{4}}
-\newlabel{thm:edge-deletion-4colorable}{{7.2}{4}}
-\newlabel{thm:edge-deletion-coloring-structure}{{7.3}{4}}
-\gdef \@abspage@last{4}

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papers/flip_symmetric_maximal_planar_graphs/paper.tex

@@ -276,80 +276,6 @@ 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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