From 4693f632088d5e38cdf1601ae47305352572fa97 Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 22 May 2026 12:24:07 -0400 Subject: [PATCH] Add small-n ELG enumeration table to even_level_graph_generators 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) --- papers/even_level_graph_generators/paper.aux | 23 +- .../paper.fdb_latexmk | 35 +-- papers/even_level_graph_generators/paper.fls | 148 +---------- papers/even_level_graph_generators/paper.log | 247 ++++++++++++------ papers/even_level_graph_generators/paper.out | 5 +- papers/even_level_graph_generators/paper.pdf | Bin 1093752 -> 1098280 bytes papers/even_level_graph_generators/paper.tex | 45 ++++ 7 files changed, 235 insertions(+), 268 deletions(-) diff --git a/papers/even_level_graph_generators/paper.aux b/papers/even_level_graph_generators/paper.aux index d8e08dc..62baa8d 100644 --- a/papers/even_level_graph_generators/paper.aux +++ b/papers/even_level_graph_generators/paper.aux @@ -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} \newlabel{tocindent3}{0pt} -\@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} diff --git a/papers/even_level_graph_generators/paper.fdb_latexmk b/papers/even_level_graph_generators/paper.fdb_latexmk index 93cd83b..e603040 100644 --- a/papers/even_level_graph_generators/paper.fdb_latexmk +++ b/papers/even_level_graph_generators/paper.fdb_latexmk @@ -1,25 +1,6 @@ # Fdb version 3 -["pdflatex"] 1779464918 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779464920 - "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779464918 21409 f57d3b5341eb97db9586774d14c6260d "" - "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" - 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