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Add figures, Kempe-cycle section, and restriction experiments

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Adds two TikZ figures (boundary-state worst cases and annular cycle
counterexample), a new subsection on Kempe-cycle conservation across
medial tires, and the experiment scripts/findings for the medial tire
restriction search and annular cycle condition check.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-06-11 01:16:05 -04:00
parent 20fe6c24ca
commit 4062e87c61
10 changed files with 1418 additions and 239 deletions
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papers/medial_tire_decompositions_of_plane_triangulations/paper.aux
+14 -9
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@@ -15,20 +15,25 @@
\newlabel{def:medial-restriction-relation}{{3.5}{3}}
\citation{bauerfeld-nested-tire-decompositions}
\citation{bauerfeld-nested-tire-decompositions}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Three six-face full medial tire graphs found by the boundary-state restriction search. Black vertices are annular medial vertices; blue vertices are outer boundary medial vertices and red vertices are inner boundary medial vertices. The word below each diagram records the outer/inner type of the six annular faces in cyclic order. Boundary states are identified only up to colour permutation, not by rotation or reflection of the boundary order.}}{4}{}\protected@file@percent }
\newlabel{fig:medial-restriction-worst-cases}{{1}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A proper vertex $3$-colouring of the full medial graph of the first seven-vertex counterexample found by the experiment. The medial vertex labelled $ij$ corresponds to the edge $(i,j)$ of the triangulation. For the vertex-source decomposition at source $1$, the highlighted annular medial cycle has colour counts $(2,2,2)$, so it is not coloured with two colours except at at most one vertex.}}{4}{}\protected@file@percent }
\newlabel{fig:medial-annular-cycle-counterexample}{{2}{4}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
\newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
\newlabel{def:compatible-family}{{4.2}{4}}
\newlabel{prop:gluing-criterion}{{4.3}{4}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{4}{}\protected@file@percent }
\newlabel{def:compatible-family}{{4.2}{5}}
\newlabel{prop:gluing-criterion}{{4.3}{5}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{5}{}\protected@file@percent }
\newlabel{def:medial-boundary-state}{{5.1}{5}}
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
\newlabel{conj:medial-route-fct}{{5.3}{6}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{6}{}\protected@file@percent }
\bibcite{bauerfeld-nested-tire-decompositions}{1}
\bibcite{tait-original}{2}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent2}{29.38873pt}
\newlabel{tocindent3}{0pt}
\newlabel{def:medial-boundary-state}{{5.1}{5}}
\newlabel{conj:medial-chain-pigeonhole}{{5.2}{5}}
\newlabel{conj:medial-route-fct}{{5.3}{5}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{5}{}\protected@file@percent }
\gdef \@abspage@last{5}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{}\protected@file@percent }
\gdef \@abspage@last{8}