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Add bridge-derived census (n=6..10) to the disjunction section

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Cross-tabulate bridge-derived vs intertwining-tree coverage: the
bridge-derived share falls from 100% (n=6) to 62.7% (n=10), the
disjunction never relies on it alone, and the "neither" column is
identically zero throughout.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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didericis
2026-06-19 02:27:49 -04:00
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commit d9007c8697
7 changed files with 191 additions and 49 deletions
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papers/even_level_graph_generators/paper.aux
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@@ -44,19 +44,21 @@
\newlabel{def:intertwining-tree}{{4.6}{7}{Intertwining tree}{theorem.4.6}{}}
\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{7}{}{theorem.4.7}{}}
\newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here.}}{8}{table.2}\protected@file@percent }
\newlabel{tab:bridge-census}{{2}{8}{Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here}{table.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{8}{section*.2}\protected@file@percent }
\citation{holton-mckay}
\@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{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{8}{section*.3}\protected@file@percent }
\@writefile{lot}{\contentsline {table}{\numberline {3}{\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.}}{9}{table.3}\protected@file@percent }
\newlabel{tab:n21}{{3}{9}{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.3}{}}
\@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.}}{9}{figure.5}\protected@file@percent }
\newlabel{fig:n21-duals}{{5}{9}{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}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{10}{figure.6}\protected@file@percent }
\newlabel{fig:n24-dual}{{6}{10}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{10}{section*.3}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Beyond $n = 24$: enumeration and the next $5$-connected core}}{10}{section*.4}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{11}{section*.5}\protected@file@percent }
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{11}{figure.6}\protected@file@percent }
\newlabel{fig:n24-dual}{{6}{11}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The $25$-vertex dual $T_{25}$ of the unique $46$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only such cubic graph at $46$ vertices and the second internally $6$-connected core known. Drawn crossing-free and coloured by parity (blue even, orange odd) for its witness partition. $T_{25}$ is internally $6$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even parity subgraph.}}{12}{figure.7}\protected@file@percent }
\newlabel{fig:n25-dual}{{7}{12}{The $25$-vertex dual $T_{25}$ of the unique $46$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only such cubic graph at $46$ vertices and the second internally $6$-connected core known. Drawn crossing-free and coloured by parity (blue even, orange odd) for its witness partition. $T_{25}$ is internally $6$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even parity subgraph}{figure.7}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{12}{section*.5}\protected@file@percent }
\bibcite{holton-mckay}{1}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{14.69437pt}