From 2158c541173d3b9f21630f94706cc58aa561b78c Mon Sep 17 00:00:00 2001 From: didericis Date: Sun, 24 May 2026 14:35:35 -0400 Subject: [PATCH] dual_decomposition: abstract + intro, rename to "edge suppression", bib - Fill in the abstract and add a Section 1 introduction. - Rename "cubic-graph edge contraction" -> "edge suppression" throughout (section, definition, theorem, captions, prose, labels). The PNG filenames keep their old paths and still resolve. - Reframe edge suppression as a classical operation we recall, not a new concept we introduce; the face-monochromatic-pair conjecture (with its strengthening) is the sole contribution. - Add a bibliography citing Appel-Haken Parts I and II, Robertson-Sanders- Seymour-Thomas, and Gonthier, and \cite them in the intro. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 76 +++--- .../paper.log | 119 +++++----- .../paper.pdf | Bin 941265 -> 1015233 bytes .../paper.tex | 219 +++++++++++++----- 4 files changed, 271 insertions(+), 143 deletions(-) diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.aux b/papers/dual_decomposition_minimal_counterexamples/paper.aux index f615b7e..3a68a03 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.aux +++ b/papers/dual_decomposition_minimal_counterexamples/paper.aux @@ -1,35 +1,51 @@ \relax -\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The minimal counterexample}}{1}{}\protected@file@percent } -\newlabel{lem:triangulate}{{1.1}{1}} -\newlabel{def:minimal}{{1.2}{1}} -\newlabel{lem:mindeg}{{1.4}{1}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The reduced dual}}{2}{}\protected@file@percent } -\newlabel{def:reduced-dual}{{2.1}{2}} -\newlabel{def:edge-names}{{2.3}{2}} -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 2.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{3}{}\protected@file@percent } -\newlabel{fig:reduced-dual-steps}{{1}{3}} -\newlabel{lem:pentagonal-externals}{{2.4}{3}} -\newlabel{lem:chord-apex}{{2.6}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 2.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 2.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{5}{}\protected@file@percent } -\newlabel{fig:chord-apex-proof}{{2}{5}} -\newlabel{lem:kempe-spike}{{2.7}{6}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Cubic-graph edge contraction}}{7}{}\protected@file@percent } -\newlabel{def:cubic-edge-contraction}{{3.1}{7}} -\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.1\hbox {}). Left: a fragment of a cubic plane graph with the contracted edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. 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The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the contraction).}}{9}{}\protected@file@percent } -\newlabel{fig:thm-cubic-contraction-4face}{{4}{9}} -\newlabel{rem:conj-3-6-empirical}{{4.2}{9}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{4.3}{10}} -\newlabel{rem:conj-3-8-empirical}{{4.4}{10}} +\citation{AH77a} +\citation{AHK77} +\citation{RSST97} +\citation{Gonthier08} +\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent } +\newlabel{sec:intro}{{1}{1}} +\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Organization.}}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Companion paper.}}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The minimal counterexample}}{2}{}\protected@file@percent } +\newlabel{sec:minimal}{{2}{2}} +\newlabel{lem:triangulate}{{2.1}{2}} +\newlabel{def:minimal}{{2.2}{2}} +\newlabel{lem:mindeg}{{2.4}{3}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The reduced dual}}{3}{}\protected@file@percent } +\newlabel{sec:reduced-dual}{{3}{3}} +\newlabel{def:reduced-dual}{{3.1}{3}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{4}{}\protected@file@percent } +\newlabel{fig:reduced-dual-steps}{{1}{4}} +\newlabel{def:edge-names}{{3.3}{4}} +\newlabel{lem:pentagonal-externals}{{3.4}{4}} +\newlabel{lem:chord-apex}{{3.6}{5}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 3.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{6}{}\protected@file@percent } +\newlabel{fig:chord-apex-proof}{{2}{6}} +\newlabel{lem:kempe-spike}{{3.7}{7}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Edge suppression}}{8}{}\protected@file@percent } +\newlabel{sec:edge-suppression}{{4}{8}} +\newlabel{def:edge-suppression}{{4.1}{8}} +\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{8}{}\protected@file@percent } +\newlabel{fig:edge-suppression}{{3}{8}} +\newlabel{thm:edge-suppression-4face}{{4.2}{8}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{9}{}\protected@file@percent } +\newlabel{sec:toward-4ct}{{5}{9}} +\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{9}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the suppressed graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the suppression).}}{10}{}\protected@file@percent } +\newlabel{fig:thm-edge-suppression-4face}{{4}{10}} +\newlabel{rem:conj-3-6-empirical}{{5.2}{10}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{5.3}{11}} +\newlabel{rem:conj-3-8-empirical}{{5.4}{11}} +\bibcite{AH77a}{1} +\bibcite{AHK77}{2} +\bibcite{RSST97}{3} +\bibcite{Gonthier08}{4} \newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{0pt} +\newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\newlabel{rem:implication-4ct}{{4.5}{11}} -\gdef \@abspage@last{11} +\newlabel{rem:implication-4ct}{{5.5}{12}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{12}{}\protected@file@percent } +\gdef \@abspage@last{12} diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.log b/papers/dual_decomposition_minimal_counterexamples/paper.log index 2761a81..9c74678 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.log +++ b/papers/dual_decomposition_minimal_counterexamples/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) 24 MAY 2026 14:15 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:32 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -192,121 +192,128 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv e )) [1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -Overfull \hbox (41.917pt too wide) in paragraph at lines 145--147 +[2] +Overfull \hbox (41.917pt too wide) in paragraph at lines 233--235 []\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 = (\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$. 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