From 6c7bd9e0db0b7cf03e5de294fb397bd999c8a93f Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 00:31:40 -0400 Subject: [PATCH] Add Kempe-chain structure of 4-colorings of edge-deletion subgraphs For G_0 a minimum-order 5-chromatic maximal planar graph and any 4-coloring of G_0 - uv, the endpoints u, v must share a color, and the color classes pairing that color with each of two other colors must each induce a u-v path. The Kempe-chain parts follow from a standard swap-on-component contradiction against the shared-color claim. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 1 + .../paper.fdb_latexmk | 6 ++-- .../paper.log | 12 +++---- .../paper.pdf | Bin 186074 -> 189743 bytes .../paper.tex | 32 ++++++++++++++++++ 5 files changed, 42 insertions(+), 9 deletions(-) diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux index 024f1e4..e9edb24 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.aux @@ -17,4 +17,5 @@ \@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Edge-deletion subgraphs}}{4}{}\protected@file@percent } \newlabel{def:edge-deletion}{{7.1}{4}} \newlabel{thm:edge-deletion-4colorable}{{7.2}{4}} +\newlabel{thm:edge-deletion-coloring-structure}{{7.3}{4}} \gdef \@abspage@last{4} diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk index 7e53d2b..d1f118a 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1778732745 "paper.tex" "paper.pdf" "paper" 1778732746 +["pdflatex"] 1778733012 "paper.tex" "paper.pdf" "paper" 1778733012 "/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 "" @@ -122,8 +122,8 @@ "/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" 1778732746 1438 b028da99404765220b31b8b219d41cca "pdflatex" - "paper.tex" 1778732735 12434 2966cec5ecd6b41acdbce2ab2207a710 "" + "paper.aux" 1778733012 1496 56d68f0294258d5724b6317ee89e6aaa "pdflatex" + "paper.tex" 1778733006 13839 954cc3cf73e27b836fc30fd8b0d7a40b "" (generated) "paper.aux" "paper.log" diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.log b/papers/flip_symmetric_maximal_planar_graphs/paper.log index 13c28e1..7b97ae0 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.log +++ b/papers/flip_symmetric_maximal_planar_graphs/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 00:25 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 00:30 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -504,10 +504,10 @@ hs \OT1/cmr/m/n/10 with \OT1/cmtt/m/n/10 minimum[]connectivity $\OT1/cmr/m/n/10 [2] [3] [4] (./paper.aux) ) Here is how much of TeX's memory you used: - 13155 strings out of 478268 - 265572 string characters out of 5846347 - 542781 words of memory out of 5000000 - 30993 multiletter control sequences out of 15000+600000 + 13156 strings out of 478268 + 265610 string characters out of 5846347 + 542791 words of memory out of 5000000 + 30994 multiletter control sequences out of 15000+600000 475834 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s @@ -526,7 +526,7 @@ mti10.pfb> -Output written on paper.pdf (4 pages, 186074 bytes). +Output written on paper.pdf (4 pages, 189743 bytes). 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