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Introduce flip neighborhood and contradiction target Thm 4.5

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Defines the flip neighborhood N(G) and recasts the colored edge flip
class as a transitive closure rather than a single-step set, then
states Theorem 4.5: no colored flip class of a flip-neighbor of a
minimum-order 5-chromatic G contains G itself. The proof is one
inductive line from the definition; the theorem is intended as the
contradiction target for a future argument that some other condition
would force G into such a class.
This commit is contained in:
didericis
2026-05-14 03:15:12 -04:00
parent c59d2e95e1
commit 30f137aa06
5 changed files with 79 additions and 51 deletions
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papers/colored_edge_flip_classes/paper.aux
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip-symmetric maximal planar graphs}}{1}{}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces An edge flip replaces the diagonal $uv$ of the quadrilateral $uwvx$ with the diagonal $wx$.}}{2}{}\protected@file@percent }
\newlabel{def:flip-symmetric}{{3.1}{2}}
\newlabel{def:colored-flip-class}{{3.2}{2}}
\newlabel{def:flip-neighborhood}{{3.2}{2}}
\newlabel{def:colored-flip-class}{{3.3}{2}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{A minimal four-colorable counterexample}}{2}{}\protected@file@percent }
\newlabel{def:edge-deletion}{{4.1}{2}}
\newlabel{lem:edge-deletion-4colorable}{{4.2}{2}}
@@ -15,6 +16,7 @@
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{thm:min-five-chromatic-not-flip-symmetric}{{4.4}{3}}
\newlabel{thm:no-colored-class-contains-G}{{4.5}{3}}
\@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}}
\gdef \@abspage@last{4}
papers/colored_edge_flip_classes/paper.fdb_latexmk
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papers/colored_edge_flip_classes/paper.log
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papers/colored_edge_flip_classes/paper.tex
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@@ -172,19 +172,30 @@ $G^{\mathrm{flip}(uv)} \cong G$. We write $\mathcal{F}$ for the class
of flip-symmetric maximal planar graphs.
\end{definition}
\begin{definition}[Flip neighborhood]\label{def:flip-neighborhood}
Let $G$ be a maximal planar graph. The \emph{flip neighborhood} of
$G$ is the set
\[
\mathcal{N}(G) \;=\; \bigl\{\, G^{\mathrm{flip}(uv)} : uv \in E(G)
\text{ and the flip at } uv \text{ is admissible} \,\bigr\}
\]
of maximal planar graphs obtainable from $G$ by a single admissible
edge flip.
\end{definition}
\begin{definition}[Colored edge flip class]\label{def:colored-flip-class}
Let $G$ be a maximal planar graph and let $\varphi$ be a proper
$4$-coloring of $G$. The \emph{colored edge flip class} of
$(G, \varphi)$ is the set
\[
\mathcal{C}(G, \varphi) \;=\; \bigl\{\, G^{\mathrm{flip}(uv)} :
uv \in E(G),\ \text{the flip at } uv \text{ is admissible, and}\
\varphi(w) \neq \varphi(x) \,\bigr\},
\]
where $w, x$ are the third vertices of the two triangular faces of
$G$ containing $uv$. Equivalently, $\mathcal{C}(G, \varphi)$ is the
set of graphs obtained from $G$ by an admissible edge flip under
which $\varphi$ remains a proper $4$-coloring.
$(G, \varphi)$ is the set $\mathcal{C}(G, \varphi)$ of maximal planar
graphs reachable from $G$ by some (possibly empty) sequence of
admissible edge flips, each of which leaves $\varphi$ a proper
$4$-coloring of the resulting graph. Explicitly,
$H \in \mathcal{C}(G, \varphi)$ iff there exist graphs
$G = G_0, G_1, \ldots, G_k = H$ such that for each $0 \leq i < k$,
$G_{i+1} = G_i^{\mathrm{flip}(u_i v_i)}$ for some
$u_i v_i \in E(G_i)$ whose flip is admissible in $G_i$ and whose
opposite vertices $w_i, x_i$ satisfy
$\varphi(w_i) \neq \varphi(x_i)$.
\end{definition}
\section{A minimal four-colorable counterexample}
@@ -250,8 +261,7 @@ applied to $\varphi'$.
\begin{theorem}\label{thm:min-five-chromatic-not-flip-symmetric}
Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$.
Then for every edge $e \in E(G)$, the graph induced by an edge flip
of $e$ is $4$-colorable.
Then every $H \in \mathcal{N}(G)$ is $4$-colorable.
\end{theorem}
\begin{proof}
@@ -324,6 +334,27 @@ exists.}
\label{fig:flip-proof-case-two}
\end{figure}
\begin{theorem}\label{thm:no-colored-class-contains-G}
Let $G$ be a minimum-order maximal planar graph with $\chi(G) \geq 5$.
Then for every $H \in \mathcal{N}(G)$ and every proper $4$-coloring
$\varphi$ of $H$,
\[
G \;\notin\; \mathcal{C}(H, \varphi).
\]
\end{theorem}
\begin{proof}
Suppose, for contradiction, that $G \in \mathcal{C}(H, \varphi)$ for
some $H \in \mathcal{N}(G)$ and some proper $4$-coloring $\varphi$ of
$H$. By Definition~\ref{def:colored-flip-class}, there exists a
sequence of maximal planar graphs $H = H_0, H_1, \ldots, H_k = G$ in
which each $H_{i+1}$ is obtained from $H_i$ by an admissible edge
flip that leaves $\varphi$ a proper $4$-coloring of $H_{i+1}$. By
induction on $i$, $\varphi$ is a proper $4$-coloring of every $H_i$;
in particular, $\varphi$ is a proper $4$-coloring of $H_k = G$. But
$\chi(G) \geq 5$ admits no such coloring, a contradiction.
\end{proof}
\end{document}
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