From 30f137aa06ca65563a6a5d16f745f06a4b7477c4 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 03:15:12 -0400 Subject: [PATCH] Introduce flip neighborhood and contradiction target Thm 4.5 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. --- papers/colored_edge_flip_classes/paper.aux | 4 +- .../paper.fdb_latexmk | 6 +- papers/colored_edge_flip_classes/paper.log | 65 ++++++++---------- papers/colored_edge_flip_classes/paper.pdf | Bin 199834 -> 211837 bytes papers/colored_edge_flip_classes/paper.tex | 55 +++++++++++---- 5 files changed, 79 insertions(+), 51 deletions(-) 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 . 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