From 151f3793844102d3f52ada37814b173daf9ec871 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 14 May 2026 03:19:21 -0400 Subject: [PATCH] Lead motivation with the flip-neighborhood claim, not flip-symmetry The original framing presented flip-symmetry as the principal property and the stronger statement (every flip-neighbor of $G_0$ is 4-colorable) as a parenthetical. Reverse the emphasis: lead with the stronger claim, derive flip-asymmetry as a corollary, then introduce the colored edge flip class and Theorem 4.5 to preview the fine-grained per-coloring version of the same rigidity. Co-Authored-By: Claude Opus 4.7 --- papers/colored_edge_flip_classes/paper.aux | 4 +-- .../paper.fdb_latexmk | 6 ++-- papers/colored_edge_flip_classes/paper.log | 12 +++---- papers/colored_edge_flip_classes/paper.pdf | Bin 212852 -> 213968 bytes papers/colored_edge_flip_classes/paper.tex | 32 +++++++++++++----- 5 files changed, 32 insertions(+), 22 deletions(-) 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 . 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