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Update LaTeX build artifacts

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Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-22 11:28:18 -04:00
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3 changed files with 29 additions and 39 deletions
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papers/even_level_graph_generators/paper.aux
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@@ -40,10 +40,9 @@
\newlabel{def:intertwining-tree}{{4.6}{4}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{4}{}{theorem.4.7}{}}
\citation{holton-mckay}
\newlabel{conj:every-triangulation-derived}{{4.8}{5}{}{theorem.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Empirical status}}{5}{section*.1}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{5}{section*.2}\protected@file@percent }
\bibcite{holton-mckay}{1}
\newlabel{conj:every-triangulation-derived}{{4.8}{5}{}{theorem.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{5}{section*.1}\protected@file@percent }
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@@ -51,9 +50,7 @@
\newlabel{tocindent3}{0pt}
\@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 }
\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}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{6}{section*.3}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The witness Even Level Graph for each of the six Holton--McKay duals, drawn as a crossing-free planar graph and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The dashed red edges are the same-parity edges that the bridge switches flip; flipping them yields the corresponding dual in Figure\nonbreakingspace \ref {fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs outright, so no edge is flipped.}}{7}{figure.5}\protected@file@percent }
\newlabel{fig:n21-elgs}{{5}{7}{The witness Even Level Graph for each of the six Holton--McKay duals, drawn as a crossing-free planar graph and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The dashed red edges are the same-parity edges that the bridge switches flip; flipping them yields the corresponding dual in Figure~\ref {fig:n21-duals}. Duals $1$ and $2$ are Even Level Graphs outright, so no edge is flipped}{figure.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs with the same parity colouring. The solid green edges are the bridge edges introduced by the switches from the Even Level Graphs of Figure\nonbreakingspace \ref {fig:n21-elgs}. 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.}}{8}{figure.6}\protected@file@percent }
\newlabel{fig:n21-duals}{{6}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs with the same parity colouring. The solid green edges are the bridge edges introduced by the switches from the Even Level Graphs of Figure~\ref {fig:n21-elgs}. 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.6}{}}
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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 }
\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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