From bd409585baeb641e660e38098ebcbf37db97a8cf Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 00:18:34 -0400 Subject: [PATCH] Frame flip-asymmetry as first of further necessary properties Adds a transitional section reframing the frequency results: the relevant class is not all maximal planar graphs but those that resist Kempe-style reductions, where flip-asymmetry's exclusion may have real bite. Sets up subsequent development of additional necessary properties of a minimum-order 5-chromatic counterexample. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 3 +- .../paper.fdb_latexmk | 6 +-- .../paper.log | 36 +++++++++--------- .../paper.pdf | Bin 164862 -> 165847 bytes .../paper.tex | 13 +++++++ 5 files changed, 36 insertions(+), 22 deletions(-) diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.aux b/papers/flip_symmetric_maximal_planar_graphs/paper.aux index b17b3d3..0b58443 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.aux +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.aux @@ -13,4 +13,5 @@ \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\gdef \@abspage@last{3} +\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Further necessary properties of a minimal counterexample}}{3}{}\protected@file@percent } +\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 ea19a55..56970d7 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"] 1778731932 "paper.tex" "paper.pdf" "paper" 1778731932 +["pdflatex"] 1778732276 "paper.tex" "paper.pdf" "paper" 1778732276 "/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 "" @@ -120,8 +120,8 @@ "/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" 1778731932 1086 27c5b16223020a876dd8d169d7dbeb8e "pdflatex" - "paper.tex" 1778731732 10481 5b166fa0a035792f320327355568546e "" + "paper.aux" 1778732276 1234 809d17d72706986551d4267509a33a08 "pdflatex" + "paper.tex" 1778732267 11154 4367545e63b37b360cb8c3c1c287d046 "" (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 25b8e89..1530de5 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 00:12 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 14 MAY 2026 00:17 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -502,7 +502,7 @@ hs \OT1/cmr/m/n/10 with \OT1/cmtt/m/n/10 minimum[]connectivity $\OT1/cmr/m/n/10 = [] -[2] [3] (./paper.aux) ) +[2] [3] [4] (./paper.aux) ) Here is how much of TeX's memory you used: 13153 strings out of 478268 265523 string characters out of 5846347 @@ -511,23 +511,23 @@ Here is how much of TeX's memory you used: 475834 words of font info for 54 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 100i,8n,104p,386b,794s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (3 pages, 164862 bytes). + +Output written on paper.pdf (4 pages, 165847 bytes). 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zQ^!0KI&B=A<eML9DxHbpc-LNhQ!H#kB>K|w_`G%+?dIYBouK|WmyFA7X$ZfA68ATl&E zm%(5HDSsu*y)MIX6vpxA`4vUcH}6I13q{%7fD4dFL?UTYNfU!aBGJU&(SjS0m~`l7 z?cxS%U^a=wKoiM-<~+yp$@h11eg^=41CSXKaWO0?Kp+X#il~T!SQRC)F4n|`D2t-l zWN5zl-z^anfk^ayRuzL{fXKc3Y($KTgcuX!Vt+y;MM|VaMr6e#QGWCTQ$+pBYnr%u z_L?C&U9VZ<^}%b7_{n VWS6{$0uT!{Gd4L2B_%~qMhcCFaZmsN diff --git a/papers/flip_symmetric_maximal_planar_graphs/paper.tex b/papers/flip_symmetric_maximal_planar_graphs/paper.tex index 89a49e5..34fac10 100644 --- a/papers/flip_symmetric_maximal_planar_graphs/paper.tex +++ b/papers/flip_symmetric_maximal_planar_graphs/paper.tex @@ -276,6 +276,19 @@ the minimum-degree-$5$ class --- which already contains every candidate minimum-order $5$-chromatic graph --- flip-symmetric examples become a vanishing fraction. +\section{Further necessary properties of a minimal counterexample} + +The frequency data of Section~\ref{sec:frequency} look unflattering +only when flip-symmetry is weighed against the full class of maximal +planar graphs. The class that actually matters --- minimum-order +$5$-chromatic triangulations that also resist every Kempe-style +reduction --- is far thinner, and flip-symmetry may exclude a +substantially larger fraction of it if the configurations it removes +overlap those responsible for Kempe reducibility. We therefore turn +to identifying further necessary properties of a minimum-order +$5$-chromatic maximal planar graph, of which flip-asymmetry is the +first. + \end{document} %-----------------------------------------------------------------------