didericis
d99f8e23b3
- Add Definition 3.1 "Heawood number of a vertex" (+1 if CW colour order is (1,2,3), -1 if (1,3,2)) and cite Heawood 1898 in the bibliography. - Add Lemma 5.2 "Heawood number is constant on the Kempe cycles through the merged edge", positioned immediately after Conjecture 5.1. Its proof exhibits a (F, e_1, e_2) witness for clauses (1)-(3) of the conjecture from any pair (v_0, v_1) of consecutive K-vertices with differing Heawood signs, by cases on whether phi(e) = a or b. The proof does not invoke Conjecture 5.3 or Theorem 4.X. - Add a two-panel figure illustrating Case A (b-edges on F_R when phi(e) = a) and Case B (a-edges on F_L when phi(e) = b), with the cyclic colour orders (a, b, c) at v_0 and (a, c, b) at v_1 visible from the angular layout. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
58 lines
5.5 KiB
TeX
58 lines
5.5 KiB
TeX
\relax
|
|
\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}}
|
|
\citation{Heawood1898}
|
|
\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:heawood-number}{{3.1}{3}}
|
|
\newlabel{def:reduced-dual}{{3.2}{4}}
|
|
\newlabel{def:edge-names}{{3.4}{4}}
|
|
\newlabel{lem:pentagonal-externals}{{3.5}{4}}
|
|
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.2\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}$.}}{5}{}\protected@file@percent }
|
|
\newlabel{fig:reduced-dual-steps}{{1}{5}}
|
|
\newlabel{lem:chord-apex}{{3.7}{6}}
|
|
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.7\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 3.5\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent }
|
|
\newlabel{fig:chord-apex-proof}{{2}{7}}
|
|
\newlabel{lem:kempe-spike}{{3.8}{8}}
|
|
\@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.}}{9}{}\protected@file@percent }
|
|
\newlabel{fig:edge-suppression}{{3}{9}}
|
|
\newlabel{thm:edge-suppression-4face}{{4.2}{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}}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{10}{}\protected@file@percent }
|
|
\newlabel{sec:toward-4ct}{{5}{10}}
|
|
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}}
|
|
\newlabel{lem:kempe-heawood-constant}{{5.2}{11}}
|
|
\newlabel{rem:conj-3-6-empirical}{{5.3}{11}}
|
|
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with $h_\varphi (v_0) = +1$ (clockwise colour order $(a, b, c)$) and $h_\varphi (v_1) = -1$ (clockwise order $(a, c, b)$). \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the two $b$-edges at $v_0, v_1$ lie on the same face $F$, with $e$ as the unique $\partial F$-edge between them. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the two $a$-edges at $v_0, v_1$ lie on the opposite face $F$ instead, again with $e$ between them on one arc. In either case $(F, e_1, e_2)$ witnesses clauses (1)--(3) of Conjecture\nonbreakingspace 5.1\hbox {}.}}{12}{}\protected@file@percent }
|
|
\newlabel{fig:lemma-kempe-heawood}{{5}{12}}
|
|
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.4}{13}}
|
|
\newlabel{rem:conj-3-8-empirical}{{5.5}{13}}
|
|
\newlabel{rem:implication-4ct}{{5.6}{13}}
|
|
\bibcite{Heawood1898}{1}
|
|
\bibcite{AH77a}{2}
|
|
\bibcite{AHK77}{3}
|
|
\bibcite{RSST97}{4}
|
|
\bibcite{Gonthier08}{5}
|
|
\newlabel{tocindent-1}{0pt}
|
|
\newlabel{tocindent0}{12.7778pt}
|
|
\newlabel{tocindent1}{17.77782pt}
|
|
\newlabel{tocindent2}{0pt}
|
|
\newlabel{tocindent3}{0pt}
|
|
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{14}{}\protected@file@percent }
|
|
\gdef \@abspage@last{14}
|