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Draft abstract and keywords

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MSC class and author affiliation remain to be filled in.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-14 03:20:18 -04:00
parent 151f379384
commit c2e909d5b2
6 changed files with 53 additions and 36 deletions
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papers/colored_edge_flip_classes/paper.aux
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papers/colored_edge_flip_classes/paper.fdb_latexmk
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papers/colored_edge_flip_classes/paper.pdf
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papers/colored_edge_flip_classes/paper.tex
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\subjclass[2010]{Primary }
\keywords{}
\keywords{Four Color Theorem, edge flip, maximal planar graph,
Kempe chain, minimum counterexample}
\date{}
\dedicatory{}
\begin{abstract}
Suppose the Four Color Theorem fails, and let $G_0$ be a maximal
planar graph of minimum order with $\chi(G_0) \geq 5$. Using an
edge-deletion argument together with a Kempe-chain swap, we show
that every graph in the flip neighborhood $\mathcal{N}(G_0)$ --- the
set of maximal planar graphs obtainable from $G_0$ by a single
admissible edge flip --- is $4$-colorable. In particular, no such
$G_0$ is flip-symmetric. We also introduce the colored edge flip
class $\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a
proper $4$-coloring $\varphi$ of $H$, and record that
$G_0 \notin \mathcal{C}(H, \varphi)$ for any
$H \in \mathcal{N}(G_0)$ and any such $\varphi$.
\end{abstract}
\maketitle