diff --git a/papers/colored_edge_flip_classes/paper.aux b/papers/colored_edge_flip_classes/paper.aux index 86df3ab..8560cc4 100644 --- a/papers/colored_edge_flip_classes/paper.aux +++ b/papers/colored_edge_flip_classes/paper.aux @@ -4,7 +4,8 @@ \@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} diff --git a/papers/colored_edge_flip_classes/paper.fdb_latexmk b/papers/colored_edge_flip_classes/paper.fdb_latexmk index 3b35971..525a4a1 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"] 1778738215 "paper.tex" "paper.pdf" "paper" 1778738215 +["pdflatex"] 1778738955 "paper.tex" "paper.pdf" "paper" 1778738956 "/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 "" @@ -127,8 +127,8 @@ "/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" 1778738215 1634 76c9770826e2409080ea61f950bdc52f "pdflatex" - "paper.tex" 1778738201 12340 083e7cc9bfad462c72885b4568cf2fe7 "" + "paper.aux" 1778738956 1677 32d4f0477b551efb4eb21134ead928b4 "pdflatex" + "paper.tex" 1778738936 12685 245aac37998ca821ebd3d5d5a691ec64 "" (generated) "paper.aux" "paper.log" diff --git a/papers/colored_edge_flip_classes/paper.log b/papers/colored_edge_flip_classes/paper.log index 43e0bb2..77bb45a 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 01:56 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 02:27 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -496,49 +496,44 @@ cmr/bx/n/10 . []\OT1/cmr/m/n/10 A max-i-mal pla-nar graph $\OML/cmm/m/it/10 G$ [] [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -Overfull \hbox (1.3503pt too wide) detected at line 183 -\OMS/cmsy/m/n/10 C\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 G; '\OT1/cmr/m/n/10 ) = [ -] \OML/cmm/m/it/10 G[] \OT1/cmr/m/n/10 : \OML/cmm/m/it/10 uv \OMS/cmsy/m/n/10 2 - \OML/cmm/m/it/10 E\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )\OML/cmm -/m/it/10 ; []uv[] '\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 w\OT1/cmr/m/n/10 ) \OMS/c -msy/m/n/10 6\OT1/cmr/m/n/10 = \OML/cmm/m/it/10 '\OT1/cmr/m/n/10 (\OML/cmm/m/it/ -10 x\OT1/cmr/m/n/10 ) []\OML/cmm/m/it/10 ; - [] - [2] LaTeX Warning: `h' float specifier changed to `ht'. -[3] [4] (./paper.aux) ) +[3] [4] (./paper.aux) + +LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. + + ) Here is how much of TeX's memory you used: - 13205 strings out of 478268 - 266382 string characters out of 5846347 - 542802 words of memory out of 5000000 - 31040 multiletter control sequences out of 15000+600000 + 13207 strings out of 478268 + 266438 string characters out of 5846347 + 542812 words of memory out of 5000000 + 31042 multiletter control sequences out of 15000+600000 477211 words of font info for 59 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 100i,8n,104p,495b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s - -< -/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb> -Output written on paper.pdf (4 pages, 199834 bytes). + +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb> +Output written on paper.pdf (4 pages, 211837 bytes). PDF statistics: - 100 PDF objects out of 1000 (max. 8388607) - 61 compressed objects within 1 object stream + 105 PDF objects out of 1000 (max. 8388607) + 64 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 13 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/colored_edge_flip_classes/paper.pdf b/papers/colored_edge_flip_classes/paper.pdf index bcc46fd..1dbad26 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 9baf0ab..5d2d540 100644 --- a/papers/colored_edge_flip_classes/paper.tex +++ b/papers/colored_edge_flip_classes/paper.tex @@ -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} %-----------------------------------------------------------------------