diff --git a/papers/colored_edge_flip_classes/paper.aux b/papers/colored_edge_flip_classes/paper.aux index 4b27593..ceee268 100644 --- a/papers/colored_edge_flip_classes/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -1,8 +1,8 @@ \relax \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Motivation}}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Preliminaries}}{1}{}\protected@file@percent } -\@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 } +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Flip-symmetric maximal planar graphs}}{2}{}\protected@file@percent } \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}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} @@ -19,4 +18,5 @@ \newlabel{tocindent3}{0pt} \@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}} \gdef \@abspage@last{4} diff --git a/papers/colored_edge_flip_classes/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk index 161dbfb..5fff4d7 100644 --- a/papers/colored_edge_flip_classes/paper.fdb_latexmk +++ b/papers/colored_edge_flip_classes/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778743050 "paper.tex" "paper.pdf" "paper" 1778743050 +["pdflatex"] 1778743139 "paper.tex" "paper.pdf" "paper" 1778743140 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -128,8 +128,8 @@ "/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 "" "/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc "" "/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c "" - "paper.aux" 1778743050 1746 21946e155e8c5bd26d5d9b5107a85bfc "pdflatex" - "paper.tex" 1778743042 14036 c8b4296ad33d9dcc8aad020d28492369 "" + "paper.aux" 1778743139 1746 b7b4e1d574b80cd1f719787814125803 "pdflatex" + "paper.tex" 1778743123 14803 d039fbcd79ea36c0a0a4c5ba50a8e474 "" (generated) "paper.aux" "paper.log" diff --git a/papers/colored_edge_flip_classes/paper.log b/papers/colored_edge_flip_classes/paper.log index e246978..c102a31 100644 --- a/papers/colored_edge_flip_classes/paper.log +++ b/papers/colored_edge_flip_classes/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 03:17 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 03:18 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -485,17 +485,13 @@ Package epstopdf-base Info: Redefining graphics rule for `.eps' on input line 4 File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) - -LaTeX Warning: `h' float specifier changed to `ht'. - - -Overfull \hbox (6.71799pt too wide) in paragraph at lines 169--173 +[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] +Overfull \hbox (6.71799pt too wide) in paragraph at lines 183--187 []\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 [] -[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] [2] LaTeX Warning: `h' float specifier changed to `ht'. @@ -526,7 +522,7 @@ b> usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (4 pages, 212852 bytes). +Output written on paper.pdf (4 pages, 213968 bytes). PDF statistics: 105 PDF objects out of 1000 (max. 8388607) 64 compressed objects within 1 object stream diff --git a/papers/colored_edge_flip_classes/paper.pdf b/papers/colored_edge_flip_classes/paper.pdf index 243a67c..4cd66f6 100644 Binary files a/papers/colored_edge_flip_classes/paper.pdf and b/papers/colored_edge_flip_classes/paper.pdf differ diff --git a/papers/colored_edge_flip_classes/paper.tex b/papers/colored_edge_flip_classes/paper.tex index 31ed348..e938fa2 100644 --- a/papers/colored_edge_flip_classes/paper.tex +++ b/papers/colored_edge_flip_classes/paper.tex @@ -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}