Justify non-triangulated step in Lemma 4.2 contraction proof
The original proof appealed to minimality of $G_0$ to 4-color $G_0/uv$, but $G_0/uv$ is not in general a triangulation, so it is not directly covered by the minimality hypothesis (which is over maximal planar graphs). Triangulate $G_0/uv$ into a maximal planar $T$ on the same vertex set: $|V(T)| < |V(G_0)|$, so minimality gives $T$ a 4-coloring, which restricts to $G_0/uv$. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{A minimal four-colorable counterexample}}{2}{}\protected@file@percent }
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\newlabel{def:edge-deletion}{{4.1}{2}}
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\newlabel{lem:edge-deletion-4colorable}{{4.2}{2}}
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\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{2}}
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\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}}
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\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}}
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\newlabel{thm:no-colored-class-contains-G}{{4.5}{3}}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}}
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\newlabel{thm:no-colored-class-contains-G}{{4.5}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Case\nonbreakingspace 2 of the proof of Theorem\nonbreakingspace 4.4\hbox {}: $u, v$ 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.}}{4}{}\protected@file@percent }
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\newlabel{fig:flip-proof-case-two}{{2}{4}}
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@@ -215,9 +215,14 @@ $4$-colorable.
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\begin{proof}
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Fix $uv \in E(G_0)$ and let $G_0 / uv$ denote the simple planar graph
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obtained by contracting $uv$ and discarding parallel edges. Since
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$|V(G_0/uv)| = |V(G_0)| - 1$, the minimality of $G_0$ supplies a
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proper $4$-coloring $c$ of $G_0 / uv$. Let $z$ be the contracted
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obtained by contracting $uv$ and discarding parallel edges. Then
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$G_0 / uv$ is a simple planar graph on $|V(G_0)| - 1 \geq 4$ vertices
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but is not in general a triangulation; triangulate its planar
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embedding (by adding chords inside any non-triangular face) to obtain
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a maximal planar graph $T$ on the same vertex set, with $G_0 / uv$ as
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a spanning subgraph and $|V(T)| < |V(G_0)|$. By the minimality of
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$G_0$, $T$ admits a proper $4$-coloring, which restricts to a proper
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$4$-coloring $c$ of $G_0 / uv$. Let $z$ be the contracted
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vertex and define $c'\colon V(G_0) \to \{1,2,3,4\}$ by
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$c'(u) = c'(v) = c(z)$ and $c'(y) = c(y)$ for $y \notin \{u, v\}$.
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Every edge of $G_0 - uv$ is either disjoint from $\{u, v\}$ or
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