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Lead motivation with the flip-neighborhood claim, not flip-symmetry

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The original framing presented flip-symmetry as the principal
property and the stronger statement (every flip-neighbor of $G_0$ is
4-colorable) as a parenthetical.  Reverse the emphasis: lead with the
stronger claim, derive flip-asymmetry as a corollary, then introduce
the colored edge flip class and Theorem 4.5 to preview the
fine-grained per-coloring version of the same rigidity.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-14 03:19:21 -04:00
parent a856c628f2
commit 151f379384
5 changed files with 32 additions and 22 deletions
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papers/colored_edge_flip_classes/paper.aux
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\relax
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent }
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\newlabel{def:flip-symmetric}{{3.1}{2}}
\newlabel{def:flip-neighborhood}{{3.2}{2}}
\newlabel{def:colored-flip-class}{{3.3}{2}}
@@ -11,7 +11,6 @@
\newlabel{lem:edge-deletion-4colorable}{{4.2}{2}}
\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}}
\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}}
\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 }
\newlabel{fig:flip-proof-case-two}{{2}{4}}
\newlabel{thm:no-colored-class-contains-G}{{4.5}{4}}
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papers/colored_edge_flip_classes/paper.fdb_latexmk
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papers/colored_edge_flip_classes/paper.log
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[]\OT1/cmr/bx/n/10 Definition 3.1 \OT1/cmr/m/n/10 (Flip-sym-met-ric graph)\OT1/
cmr/bx/n/10 . []\OT1/cmr/m/n/10 A max-i-mal pla-nar graph $\OML/cmm/m/it/10 G$
\OT1/cmr/m/n/10 is \OT1/cmr/m/it/10 flip-symmetric
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papers/colored_edge_flip_classes/paper.tex
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@@ -91,16 +91,30 @@ property shared by every maximal planar graph $H$ with $|V(H)| =
maximal planar graphs from playing the role of a minimum
counterexample.
This paper investigates one such property: behavior under an edge
flip. Our principal observation
Our principal observation
(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that
every edge flip of a minimum-order $5$-chromatic maximal planar graph
yields a $4$-colorable graph. In particular, no such graph is
\emph{flip-symmetric}, where we call a maximal planar graph $G$
flip-symmetric when some admissible flip at an edge of $G$ returns a
graph isomorphic to $G$. The search for a counterexample to the Four
Color Theorem may therefore be confined to the complement of the
class $\mathcal{F}$ of flip-symmetric graphs.
every graph in the \emph{flip neighborhood} of $G_0$ --- the set
$\mathcal{N}(G_0)$ of maximal planar graphs obtainable from $G_0$ by
a single admissible edge flip --- is $4$-colorable. In other words,
$G_0$ sits at the boundary of $4$-colorability: a single flip in any
direction yields a $4$-colorable graph. As an immediate corollary,
no such $G_0$ is \emph{flip-symmetric}, where we call a maximal
planar graph $G$ flip-symmetric when some admissible flip at an edge
of $G$ returns a graph isomorphic to $G$; if any flip of $G_0$ were
to return $G_0$, that flip would witness $G_0$ as $4$-colorable. The
search for a counterexample to the Four Color Theorem may therefore
be confined to the complement of the class $\mathcal{F}$ of
flip-symmetric maximal planar graphs.
To track this rigidity at the level of individual $4$-colorings, we
introduce the \emph{colored edge flip class}
$\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a proper
$4$-coloring $\varphi$ of $H$: the set of maximal planar graphs
reachable from $H$ by sequences of admissible edge flips that each
preserve $\varphi$. Theorem~\ref{thm:no-colored-class-contains-G}
records that $G_0 \notin \mathcal{C}(H, \varphi)$ for any
$H \in \mathcal{N}(G_0)$ and any proper $4$-coloring $\varphi$ of
$H$.
\section{Preliminaries}