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Drop flip-symmetry framing

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Remove the flip-symmetric definition, the class $\mathcal{F}$, and
all references to flip-symmetry from the abstract, motivation, and
section 3 title.  Section 3 is renamed to reflect what remains: the
flip neighborhood and the colored edge flip class.  The principal
theorem's label is renamed to thm:flip-neighborhood-4colorable to
match its statement.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-14 03:22:33 -04:00
parent c2e909d5b2
commit 53a9192f65
5 changed files with 23 additions and 45 deletions
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papers/colored_edge_flip_classes/paper.aux
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The flip neighborhood of a minimum-order counterexample}}{3}{}\protected@file@percent } \newlabel{def:edge-deletion}{{4.1}{2}}
\newlabel{def:edge-deletion}{{4.1}{3}} \newlabel{lem:edge-deletion-4colorable}{{4.2}{2}}
\newlabel{lem:edge-deletion-4colorable}{{4.2}{3}}
\newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}} \newlabel{lem:edge-deletion-coloring-structure}{{4.3}{3}}
\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}} \newlabel{thm:flip-neighborhood-4colorable}{{4.4}{3}}
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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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@@ -81,8 +81,7 @@ planar graph of minimum order with $\chi(G_0) \geq 5$. Using an
edge-deletion argument together with a Kempe-chain swap, we show edge-deletion argument together with a Kempe-chain swap, we show
that every graph in the flip neighborhood $\mathcal{N}(G_0)$ --- the that every graph in the flip neighborhood $\mathcal{N}(G_0)$ --- the
set of maximal planar graphs obtainable from $G_0$ by a single set of maximal planar graphs obtainable from $G_0$ by a single
admissible edge flip --- is $4$-colorable. In particular, no such admissible edge flip --- is $4$-colorable. We also introduce the colored edge flip
$G_0$ is flip-symmetric. We also introduce the colored edge flip
class $\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a class $\mathcal{C}(H, \varphi)$ of a maximal planar graph $H$ and a
proper $4$-coloring $\varphi$ of $H$, and record that proper $4$-coloring $\varphi$ of $H$, and record that
$G_0 \notin \mathcal{C}(H, \varphi)$ for any $G_0 \notin \mathcal{C}(H, \varphi)$ for any
@@ -104,19 +103,12 @@ maximal planar graphs from playing the role of a minimum
counterexample. counterexample.
Our principal observation Our principal observation
(Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}) is that (Theorem~\ref{thm:flip-neighborhood-4colorable}) is that every graph
every graph in the \emph{flip neighborhood} of $G_0$ --- the set in the \emph{flip neighborhood} of $G_0$ --- the set
$\mathcal{N}(G_0)$ of maximal planar graphs obtainable from $G_0$ by $\mathcal{N}(G_0)$ of maximal planar graphs obtainable from $G_0$ by
a single admissible edge flip --- is $4$-colorable. In other words, 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 $G_0$ sits at the boundary of $4$-colorability: a single flip in any
direction yields a $4$-colorable graph. As an immediate corollary, direction yields a $4$-colorable graph.
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 To track this rigidity at the level of individual $4$-colorings, we
introduce the \emph{colored edge flip class} introduce the \emph{colored edge flip class}
@@ -182,7 +174,7 @@ is not simple and the flip is forbidden.
$uwvx$ with the diagonal $wx$.} $uwvx$ with the diagonal $wx$.}
\end{figure} \end{figure}
\section{Flip-symmetric maximal planar graphs} \section{Flip neighborhoods and colored edge flip classes}
For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$ For a maximal planar graph $G$ and an admissible edge $uv \in E(G)$
with incident triangles $uvw$, $uvx$, write with incident triangles $uvw$, $uvx$, write
@@ -191,13 +183,6 @@ with incident triangles $uvw$, $uvx$, write
\] \]
for the graph obtained from $G$ by flipping $uv$. for the graph obtained from $G$ by flipping $uv$.
\begin{definition}[Flip-symmetric graph]\label{def:flip-symmetric}
A maximal planar graph $G$ is \emph{flip-symmetric} if there exists an
admissible edge $uv \in E(G)$ such that
$G^{\mathrm{flip}(uv)} \cong G$. We write $\mathcal{F}$ for the class
of flip-symmetric maximal planar graphs.
\end{definition}
\begin{definition}[Flip neighborhood]\label{def:flip-neighborhood} \begin{definition}[Flip neighborhood]\label{def:flip-neighborhood}
Let $G$ be a maximal planar graph. The \emph{flip neighborhood} of Let $G$ be a maximal planar graph. The \emph{flip neighborhood} of
$G$ is the set $G$ is the set
@@ -290,7 +275,7 @@ applied to $\varphi'$.
(3) Identical to (2) with $c$ in place of $b$. (3) Identical to (2) with $c$ in place of $b$.
\end{proof} \end{proof}
\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric} \begin{theorem}\label{thm:flip-neighborhood-4colorable}
Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$. Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$.
Then every $H \in \mathcal{N}(G)$ is $4$-colorable. Then every $H \in \mathcal{N}(G)$ is $4$-colorable.
\end{theorem} \end{theorem}
@@ -356,7 +341,7 @@ reducing to Case~1.
\node[font=\small] at (3.5, 2.6) {$\{a, b\}$-Kempe path $P$}; \node[font=\small] at (3.5, 2.6) {$\{a, b\}$-Kempe path $P$};
\end{tikzpicture} \end{tikzpicture}
\caption{Case~2 of the proof of \caption{Case~2 of the proof of
Theorem~\ref{thm:min-five-chromatic-not-flip-symmetric}: $u, v$ share Theorem~\ref{thm:flip-neighborhood-4colorable}: $u, v$ share
color $a$ and $w, x$ share color $c$. The $\{a, b\}$-Kempe path $P$ 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 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 $\{c, d\}$-path between $w$ and $x$ can avoid crossing $P$; since the