57 lines
7.4 KiB
TeX
57 lines
7.4 KiB
TeX
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{1}{section.2}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{1}{figure.1}\protected@file@percent }
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\newlabel{fig:levels}{{1}{1}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{2}{figure.2}\protected@file@percent }
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\newlabel{fig:level-cycle}{{2}{2}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.2}{}}
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\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$.}}{2}{figure.3}\protected@file@percent }
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\newlabel{fig:edge-switch}{{3}{2}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}}{3}{figure.4}\protected@file@percent }
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\newlabel{fig:parity-subgraph}{{4}{3}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.4}{}}
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\newlabel{sec:outerplanar-components}{{3}{3}{Outerplanarity of level components}{section.3}{}}
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\newlabel{thm:outerplanar-component}{{3.1}{3}{}{theorem.3.1}{}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Even Level Graphs}}{3}{section.4}\protected@file@percent }
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\newlabel{sec:even-level-graphs}{{4}{3}{Even Level Graphs}{section.4}{}}
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\newlabel{def:even-level-graph}{{4.1}{3}{Even Level Graph}{theorem.4.1}{}}
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\newlabel{thm:even-level-4colorable}{{4.2}{3}{}{theorem.4.2}{}}
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\newlabel{def:derived-level-graph}{{4.3}{4}{Derived level graph}{theorem.4.3}{}}
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\newlabel{def:bridge-switch}{{4.4}{4}{Bridge switch}{theorem.4.4}{}}
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\newlabel{def:bridge-derived-level-graph}{{4.5}{4}{Bridge-derived level graph}{theorem.4.5}{}}
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\newlabel{def:intertwining-tree}{{4.6}{4}{Intertwining tree}{theorem.4.6}{}}
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\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{4}{}{theorem.4.7}{}}
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\citation{holton-mckay}
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\bibcite{holton-mckay}{1}
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\newlabel{conj:every-triangulation-derived}{{4.8}{5}{}{theorem.4.8}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{5}{section*.1}\protected@file@percent }
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\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{6}{table.1}\protected@file@percent }
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\newlabel{tab:n21}{{1}{6}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{6}{figure.5}\protected@file@percent }
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\newlabel{fig:n21-duals}{{5}{6}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}}
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