didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

dual_decomposition: move strengthened conjecture into Section 4 + 4CT implication

Browse Source 
- 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 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-24 13:43:09 -04:00
parent 753af5ffae
commit 440ec9cc86
4 changed files with 172 additions and 133 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/dual_decomposition_minimal_counterexamples/paper.aux
+12 -9
View File
@@ -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}