diff --git a/papers/even_level_graph_generators/paper.pdf b/papers/even_level_graph_generators/paper.pdf index 2c98eca..f790085 100644 Binary files a/papers/even_level_graph_generators/paper.pdf and b/papers/even_level_graph_generators/paper.pdf differ diff --git a/papers/even_level_graph_generators/paper.tex b/papers/even_level_graph_generators/paper.tex index 3bebd3b..2897a64 100644 --- a/papers/even_level_graph_generators/paper.tex +++ b/papers/even_level_graph_generators/paper.tex @@ -334,38 +334,6 @@ $n = 21$ and there are exactly $6$ of them. Below $n = 21$ every maximal planar graph is an intertwining tree, which is why the disjunction holds trivially in that range. -\subsection*{Empirical status} - -For each isomorphism class of maximal planar graphs on $n$ vertices, -we ask whether (i) some isomorphic representative is a bridge-derived -level graph of some Even Level Graph, and/or (ii) it is an intertwining -tree. The conjecture holds for the class iff at least one of (i), (ii) -holds. Below $n = 21$ condition (ii) holds for \emph{every} class, so -the table mainly records how far the bridge-derived disjunct (i) reaches -on its own. We classified bridge-derivability exhaustively for -$n \le 9$, where every backward bridge-orbit can be enumerated in full. - -\begin{center} -\begin{tabular}{rcccccc} -$n$ & \# iso & bridge only & inter.\ only & both & missing & status \\\hline -$6$ & $2$ & $0$ & $0$ & $2$ & $0$ & holds \\ -$7$ & $5$ & $0$ & $1$ & $4$ & $0$ & holds \\ -$8$ & $14$ & $0$ & $2$ & $12$ & $0$ & holds \\ -$9$ & $50$ & $0$ & $14$ & $36$ & $0$ & holds \\ -\end{tabular} -\end{center} - -\noindent -Here ``bridge only'' counts classes that are bridge-derived but not -intertwining trees, ``inter.\ only'' the reverse, and ``both'' the -intersection; ``missing'' counts classes that are neither (a -counterexample). The ``bridge only'' column is $0$ throughout this range -precisely because every class is an intertwining tree for $n \le 20$; -the ``inter.\ only'' counts ($1,2,14$) are the classes that the -bridge-derived disjunct alone does not yet reach, showing that -bridge-derivability is strictly weaker than ``intertwining tree'' here -and that the two disjuncts genuinely complement one another. - \subsection*{The boundary case $n = 21$} The first triangulations that are \emph{not} intertwining trees are the