From 440ec9cc8695396a04522790531513db1b4e0443 Mon Sep 17 00:00:00 2001 From: didericis Date: Sun, 24 May 2026 13:43:09 -0400 Subject: [PATCH] dual_decomposition: move strengthened conjecture into Section 4 + 4CT implication - Cut Conjecture 3.8 + Remark 3.9 from Section 3 and move into a new Section 4 "The Four Colour Theorem from a strengthened conjecture". - Add Remark 4.X spelling out the implication: clause (4)(i) forces the cyclic colour pattern (c,a,c,b) on the new 4-face f_n, two opposite edges of which satisfy the hypothesis of Theorem 3.9 verbatim; case (ii) is conjecturally reducible to case (i) via a Kempe swap on the {b,c}-cycle through X_1 X_2. Theorem 3.9 then produces the proper 3-edge-colouring of the contraction, contradicting minimality of G. - Rewrite the bridge prose into the cubic-contraction definition to reference Section 4 forward, rather than the conjecture directly. Co-Authored-By: Claude Opus 4.7 --- .../paper.aux | 21 +- .../paper.log | 101 +++++----- .../paper.pdf | Bin 1144897 -> 1148116 bytes .../paper.tex | 183 +++++++++++------- 4 files changed, 172 insertions(+), 133 deletions(-) diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.aux b/papers/dual_decomposition_minimal_counterexamples/paper.aux index b479cd5..1c3ebee 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.aux +++ b/papers/dual_decomposition_minimal_counterexamples/paper.aux @@ -23,17 +23,20 @@ \newlabel{lem:all-distinct-exists}{{3.5}{9}} \newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{3.6}{9}} \newlabel{rem:conj-3-6-empirical}{{3.7}{10}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{3.8}{10}} -\newlabel{rem:conj-3-8-empirical}{{3.9}{11}} -\newlabel{def:cubic-edge-contraction}{{3.10}{11}} -\newlabel{thm:cubic-contraction-4face}{{3.11}{11}} +\newlabel{def:cubic-edge-contraction}{{3.8}{10}} +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.8\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 $2$. 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.}}{11}{}\protected@file@percent } +\newlabel{fig:cubic-edge-contraction}{{4}{11}} +\newlabel{thm:cubic-contraction-4face}{{3.9}{11}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.9\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 contracted 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 contraction).}}{12}{}\protected@file@percent } +\newlabel{fig:thm-cubic-contraction-4face}{{5}{12}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{The Four Colour Theorem from a strengthened conjecture}}{12}{}\protected@file@percent } +\newlabel{sec:toward-4ct}{{4}{12}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{4.1}{12}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{0pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Cubic-graph edge contraction (Definition\nonbreakingspace 3.10\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 $2$. 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.}}{12}{}\protected@file@percent } -\newlabel{fig:cubic-edge-contraction}{{4}{12}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 3.11\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 contracted 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 contraction).}}{13}{}\protected@file@percent } -\newlabel{fig:thm-cubic-contraction-4face}{{5}{13}} -\gdef \@abspage@last{13} +\newlabel{rem:conj-3-8-empirical}{{4.2}{13}} +\newlabel{rem:implication-4ct}{{4.3}{13}} +\gdef \@abspage@last{14} diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.log b/papers/dual_decomposition_minimal_counterexamples/paper.log index 0f94032..a10361e 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 13:27 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 13:42 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -299,71 +299,64 @@ Underfull \hbox (badness 6094) in paragraph at lines 498--498 [7] [8 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>] [9] [10] -Underfull \hbox (badness 1648) in paragraph at lines 705--710 -\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 3.9\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T + +File: fig_cubic_edge_contraction.png Graphic file (type png) + +Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 702 +. +(pdftex.def) Requested size: 341.9989pt x 73.08138pt. + [11 <./fig_cubic_edge_contraction.png>] + +File: fig_thm_cubic_contraction_4face.png Graphic file (type png) + +Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin +e 777. +(pdftex.def) Requested size: 352.79846pt x 160.78339pt. + [12 <./fig_thm_cubic_contraction_4face.png>] +Underfull \hbox (badness 1648) in paragraph at lines 837--842 +\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 4.2\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T he strength-ened con-jec-ture was tested on the same chord- [] -Underfull \hbox (badness 1014) in paragraph at lines 705--710 +Underfull \hbox (badness 1014) in paragraph at lines 837--842 \OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 3.7[]; for each colour-ing we sought any [] - -File: fig_cubic_edge_contraction.png Graphic file (type png) - -Package pdftex.def Info: fig_cubic_edge_contraction.png used on input line 769 -. -(pdftex.def) Requested size: 341.9989pt x 73.08138pt. - -LaTeX Warning: `h' float specifier changed to `ht'. - -[11] - -File: fig_thm_cubic_contraction_4face.png Graphic file (type png) - -Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin -e 844. -(pdftex.def) Requested size: 352.79846pt x 160.78339pt. - - -LaTeX Warning: `h' float specifier changed to `ht'. - -[12 <./fig_cubic_edge_contraction.png>] [13 <./fig_thm_cubic_contraction_4face. -png>] (./paper.aux) ) +[13] [14] (./paper.aux) ) Here is how much of TeX's memory you used: - 3112 strings out of 478268 - 44739 string characters out of 5846347 - 347356 words of memory out of 5000000 - 21143 multiletter control sequences out of 15000+600000 + 3114 strings out of 478268 + 44776 string characters out of 5846347 + 346374 words of memory out of 5000000 + 21145 multiletter control sequences out of 15000+600000 478077 words of font info for 62 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,9n,76p,1306b,398s stack positions out of 10000i,1000n,20000p,200000b,200000s - -Output written on paper.pdf (13 pages, 1144897 bytes). + 69i,9n,76p,1306b,326s stack positions out of 10000i,1000n,20000p,200000b,200000s +< +/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb> +Output written on paper.pdf (14 pages, 1148116 bytes). 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