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

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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>
This commit is contained in:
didericis
2026-05-14 00:27:11 -04:00
parent bd409585ba
commit f6144b98b5
6 changed files with 53 additions and 15 deletions
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papers/flip_symmetric_maximal_planar_graphs/paper.aux
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papers/flip_symmetric_maximal_planar_graphs/paper.pdf
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papers/flip_symmetric_maximal_planar_graphs/paper.tex
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$5$-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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