51 lines
4.1 KiB
TeX
51 lines
4.1 KiB
TeX
\relax
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\citation{bauerfeld-nested-tire-decompositions}
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\citation{bauerfeld-nested-tire-decompositions}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background}}{1}{}\protected@file@percent }
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\citation{bauerfeld-nested-tire-decompositions}
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\citation{bauerfeld-nested-tire-decompositions}
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\newlabel{def:medial-graph}{{2.1}{2}}
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\newlabel{prop:medial-dual-invariance}{{2.3}{2}}
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\newlabel{cor:tait-medial}{{2.4}{2}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Medial tire pieces}}{2}{}\protected@file@percent }
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\newlabel{def:full-medial-tire}{{3.1}{2}}
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\newlabel{thm:annular-medial-colour-bound}{{3.3}{3}}
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\newlabel{def:annular-teeth}{{3.4}{3}}
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\citation{bauerfeld-nested-tire-decompositions}
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\citation{bauerfeld-nested-tire-decompositions}
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\newlabel{rem:teeth-sharing}{{3.5}{4}}
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\newlabel{rem:up-teeth-count}{{3.6}{4}}
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\newlabel{def:bite}{{3.7}{4}}
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\newlabel{rem:bite-face-count}{{3.8}{4}}
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\newlabel{def:boundary-medial-vertices}{{3.9}{4}}
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\newlabel{def:medial-restriction-relation}{{3.10}{4}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Decomposition}}{4}{}\protected@file@percent }
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\newlabel{cor:medial-tire-decomposition}{{4.1}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces A simple medial tire graph $\mathsf {M}(T)$ illustrating the tooth terminology. The thick cycle is the annular medial cycle $A(T)$, whose black vertices are the annular medial vertices. Each edge of $A(T)$ carries one tooth: up teeth (blue apexes, outer-boundary medial vertices) point into the outer region, and down teeth (red apexes, inner-boundary medial vertices) point into the inner region. The two down teeth meeting at the central shared apex (larger red vertex) form a bite; that shared apex splits the inner region into two faces, one with four down teeth on its boundary and one with none.}}{5}{}\protected@file@percent }
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\newlabel{fig:medial-teeth-example}{{1}{5}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Three six-face simple 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.}}{5}{}\protected@file@percent }
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\newlabel{fig:medial-restriction-worst-cases}{{2}{5}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\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.}}{6}{}\protected@file@percent }
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\newlabel{fig:medial-annular-cycle-counterexample}{{3}{6}}
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\newlabel{def:compatible-family}{{4.2}{6}}
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\newlabel{prop:gluing-criterion}{{4.3}{6}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{A medial pigeonhole programme}}{6}{}\protected@file@percent }
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\newlabel{def:medial-boundary-state}{{5.1}{7}}
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\newlabel{conj:medial-chain-pigeonhole}{{5.2}{7}}
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\newlabel{conj:medial-route-fct}{{5.3}{7}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Kempe-cycle conservation across medial tires}}{7}{}\protected@file@percent }
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\newlabel{lem:kempe-cycles}{{5.5}{7}}
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\newlabel{lem:kempe-conservation}{{5.6}{8}}
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\newlabel{def:kempe-balanced}{{5.7}{8}}
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\newlabel{rem:kempe-balance-necessary}{{5.8}{9}}
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\bibcite{bauerfeld-nested-tire-decompositions}{1}
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\bibcite{tait-original}{2}
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