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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The combined assignment is a proper $4$-coloring of $G$. \end{proof} +\subsection*{Enumeration for small $n$} + +Even Level Graphs are scarce among triangulations. For each $n$ we +enumerated all iso classes of plane triangulations and tested, for every +choice of source vertex, whether all level subgraphs are bipartite +(equivalently, whether every level cycle is even). +Table~\ref{tab:elg-counts} records, for $4 \leq n \leq 11$: the number of +triangulation iso classes; how many of them admit at least one Even Level +Graph source; the number of Even Level Graph iso classes (pairs $(G, S)$ +up to isomorphism, i.e.\ valid sources counted up to $\mathrm{Aut}(G)$); +and the number of \emph{flag-rooted} Even Level Graphs, +\[ + \sum_{G} \frac{4E}{|\mathrm{Aut}(G)|}\, s(G), + \qquad E = 3n - 6, +\] +where $s(G)$ is the number of valid sources of $G$. The flag-rooting is the +automorphism-free count: $\mathrm{Aut}(G)$ acts freely on the $4E$ flags of a +$3$-connected triangulation, so every summand is an integer. + +The smallest Even Level Graph is the octahedron at $n = 6$: from any vertex +the four neighbours form a $4$-cycle at level $1$ and the antipode sits alone +at level $2$. Below $n = 6$ every triangulation forces an odd level cycle, so +no Even Level Graph exists. + +\begin{table}[ht] +\centering +\begin{tabular}{ccccc} +$n$ & triangulations & with ELG source & ELG iso classes & flag-rooted ELGs \\\hline +$4$ & $1$ & $0$ & $0$ & $0$ \\ +$5$ & $1$ & $0$ & $0$ & $0$ \\ +$6$ & $2$ & $1$ & $1$ & $6$ \\ +$7$ & $5$ & $2$ & $2$ & $45$ \\ +$8$ & $14$ & $5$ & $6$ & $186$ \\ +$9$ & $50$ & $13$ & $14$ & $651$ \\ +$10$ & $233$ & $37$ & $45$ & $2766$ \\ +$11$ & $1249$ & $129$ & $169$ & $14346$ \\ +\end{tabular} +\caption{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 +$\sum_G \tfrac{4E}{|\mathrm{Aut}(G)|}\,s(G)$.} +\label{tab:elg-counts} +\end{table} + \begin{definition}[Derived level graph] \label{def:derived-level-graph} Let $G$ be an Even Level Graph with level source $S$, and let $E$ and