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) <noreply@anthropic.com>
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@@ -34,23 +34,26 @@
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\newlabel{sec:even-level-graphs}{{4}{4}{Even Level Graphs}{section.4}{}}
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\newlabel{def:even-level-graph}{{4.1}{4}{Even Level Graph}{theorem.4.1}{}}
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\newlabel{thm:even-level-4colorable}{{4.2}{4}{}{theorem.4.2}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Enumeration for small $n$}}{5}{section*.1}\protected@file@percent }
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\@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 }
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\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}{}}
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\newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}}
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\newlabel{def:bridge-switch}{{4.4}{5}{Bridge switch}{theorem.4.4}{}}
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\newlabel{def:bridge-derived-level-graph}{{4.5}{5}{Bridge-derived level graph}{theorem.4.5}{}}
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\newlabel{def:intertwining-tree}{{4.6}{5}{Intertwining tree}{theorem.4.6}{}}
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\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{5}{}{theorem.4.7}{}}
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\citation{holton-mckay}
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\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}}
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\newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}}
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\newlabel{def:intertwining-tree}{{4.6}{6}{Intertwining tree}{theorem.4.6}{}}
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\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{6}{}{theorem.4.7}{}}
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\newlabel{conj:every-triangulation-derived}{{4.8}{6}{}{theorem.4.8}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{6}{section*.1}\protected@file@percent }
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\bibcite{holton-mckay}{1}
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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.}}{7}{table.1}\protected@file@percent }
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\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}{}}
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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.}}{7}{figure.5}\protected@file@percent }
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\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}{}}
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\@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 }
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\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}{}}
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\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}{}}
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\BOOKMARK [1][-]{section*.3}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 7
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@@ -310,6 +310,51 @@ $\{\text{yellow}, \text{green}\}$. The combined assignment is a
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proper $4$-coloring of $G$.
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\end{proof}
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\subsection*{Enumeration for small $n$}
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Even Level Graphs are scarce among triangulations. For each $n$ we
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enumerated all iso classes of plane triangulations and tested, for every
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choice of source vertex, whether all level subgraphs are bipartite
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(equivalently, whether every level cycle is even).
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Table~\ref{tab:elg-counts} records, for $4 \leq n \leq 11$: the number of
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triangulation iso classes; how many of them admit at least one Even Level
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Graph source; the number of Even Level Graph iso classes (pairs $(G, S)$
|
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up to isomorphism, i.e.\ valid sources counted up to $\mathrm{Aut}(G)$);
|
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and the number of \emph{flag-rooted} Even Level Graphs,
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\[
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\sum_{G} \frac{4E}{|\mathrm{Aut}(G)|}\, s(G),
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\qquad E = 3n - 6,
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\]
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where $s(G)$ is the number of valid sources of $G$. The flag-rooting is the
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automorphism-free count: $\mathrm{Aut}(G)$ acts freely on the $4E$ flags of a
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$3$-connected triangulation, so every summand is an integer.
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The smallest Even Level Graph is the octahedron at $n = 6$: from any vertex
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the four neighbours form a $4$-cycle at level $1$ and the antipode sits alone
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at level $2$. Below $n = 6$ every triangulation forces an odd level cycle, so
|
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no Even Level Graph exists.
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\begin{table}[ht]
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\centering
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\begin{tabular}{ccccc}
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$n$ & triangulations & with ELG source & ELG iso classes & flag-rooted ELGs \\\hline
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$4$ & $1$ & $0$ & $0$ & $0$ \\
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$5$ & $1$ & $0$ & $0$ & $0$ \\
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$6$ & $2$ & $1$ & $1$ & $6$ \\
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$7$ & $5$ & $2$ & $2$ & $45$ \\
|
||||
$8$ & $14$ & $5$ & $6$ & $186$ \\
|
||||
$9$ & $50$ & $13$ & $14$ & $651$ \\
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$10$ & $233$ & $37$ & $45$ & $2766$ \\
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$11$ & $1249$ & $129$ & $169$ & $14346$ \\
|
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\end{tabular}
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\caption{Even Level Graph counts for $4 \leq n \leq 11$. The
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\emph{triangulations} column is the number of plane-triangulation iso classes
|
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(OEIS A000109). \emph{ELG iso classes} counts pairs $(G, S)$ up to
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isomorphism; \emph{flag-rooted ELGs} is the automorphism-free count
|
||||
$\sum_G \tfrac{4E}{|\mathrm{Aut}(G)|}\,s(G)$.}
|
||||
\label{tab:elg-counts}
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||||
\end{table}
|
||||
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||||
\begin{definition}[Derived level graph]
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||||
\label{def:derived-level-graph}
|
||||
Let $G$ be an Even Level Graph with level source $S$, and let $E$ and
|
||||
|
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Reference in New Issue
Block a user