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Add small-n ELG enumeration table to even_level_graph_generators

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Records, for 4<=n<=11, triangulation iso classes, how many admit an ELG
source, ELG iso classes, and the automorphism-free flag-rooted count
sum_G 4E/|Aut(G)| * s(G). Computed by experiments/count_elgs.py.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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didericis
2026-05-22 12:24:07 -04:00
parent 8fde9494d8
commit 4693f63208
7 changed files with 235 additions and 268 deletions
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papers/even_level_graph_generators/paper.aux
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@@ -34,23 +34,26 @@
\newlabel{sec:even-level-graphs}{{4}{4}{Even Level Graphs}{section.4}{}}
\newlabel{def:even-level-graph}{{4.1}{4}{Even Level Graph}{theorem.4.1}{}}
\newlabel{thm:even-level-4colorable}{{4.2}{4}{}{theorem.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{section*.1}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\DOTSB \sum@ \slimits@ _G \genfrac {}{}{}1{4E}{|\mathrm {Aut}(G)|}\,s(G)$.}}{5}{table.1}\protected@file@percent }
\newlabel{tab:elg-counts}{{1}{5}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}}
\newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}}
\newlabel{def:bridge-switch}{{4.4}{5}{Bridge switch}{theorem.4.4}{}}
\newlabel{def:bridge-derived-level-graph}{{4.5}{5}{Bridge-derived level graph}{theorem.4.5}{}}
\newlabel{def:intertwining-tree}{{4.6}{5}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{5}{}{theorem.4.7}{}}
\citation{holton-mckay}
\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}}
\newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}}
\newlabel{def:intertwining-tree}{{4.6}{6}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{6}{}{theorem.4.7}{}}
\newlabel{conj:every-triangulation-derived}{{4.8}{6}{}{theorem.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{6}{section*.1}\protected@file@percent }
\bibcite{holton-mckay}{1}
\@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.}}{7}{table.1}\protected@file@percent }
\newlabel{tab:n21}{{1}{7}{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{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.}}{7}{figure.5}\protected@file@percent }
\newlabel{fig:n21-duals}{{5}{7}{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}{}}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{14.69437pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{0pt}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{section*.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{section*.3}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {2}{\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.}}{8}{table.2}\protected@file@percent }
\newlabel{tab:n21}{{2}{8}{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.2}{}}
\@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.}}{8}{figure.5}\protected@file@percent }
\newlabel{fig:n21-duals}{{5}{8}{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}{}}
\gdef \@abspage@last{8}