diff --git a/papers/even_level_graph_generators/experiments/recompute_table.py b/papers/even_level_graph_generators/experiments/recompute_table.py new file mode 100644 index 0000000..5e40cc7 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/recompute_table.py @@ -0,0 +1,37 @@ +"""Recompute the empirical table for the bridge-derived disjunction: +for each n, count iso classes that are bridge-derived only / intertwining +only / both / neither (missing).""" +import sys +import os +sys.path.insert(0, '/Users/didericis/Code/math-research/papers/' + 'level_resolutions_of_maximal_planar_graphs/experiments') +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) +from triangulation_gen import enumerate_all_triangulations +from small_n_probe import is_bridge_derived +from test_disjunction import is_intertwining_tree + + +def main(ns): + print('n iso bridge_only inter_only both missing', flush=True) + for n in ns: + tris = enumerate_all_triangulations(n) + bo = io = both = miss = 0 + for G in tris: + bd = is_bridge_derived(G) + it = is_intertwining_tree(G) + if isinstance(it, tuple): + it = it[0] + if bd and it: + both += 1 + elif bd: + bo += 1 + elif it: + io += 1 + else: + miss += 1 + print(f'{n} {len(tris)} {bo} {io} {both} {miss}', flush=True) + + +if __name__ == '__main__': + ns = [int(x) for x in sys.argv[1:]] or [6, 7, 8, 9, 10, 11, 12] + main(ns) diff --git a/papers/even_level_graph_generators/paper.pdf b/papers/even_level_graph_generators/paper.pdf index 7b8543f..1b5d055 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 11b6815..62c0bdc 100644 --- a/papers/even_level_graph_generators/paper.tex +++ b/papers/even_level_graph_generators/paper.tex @@ -316,10 +316,15 @@ likewise $G[B]$. \begin{conjecture} \label{conj:every-triangulation-derived} -Every maximal planar graph is a valid derived level graph of some Even +Every maximal planar graph is a bridge-derived level graph of some Even Level Graph, an intertwining tree, or both. \end{conjecture} +Since a bridge-derived level graph is automatically a valid derived level +graph, this is a stronger statement than the corresponding conjecture +phrased with arbitrary $E/O$ switches; it is also the form that the +evidence below actually supports. + By Theorem~\ref{thm:intertwining-iff-hamiltonian-dual}, the intertwining-tree disjunct fails for $G$ exactly when $G^\ast$ is a counterexample to Tait's conjecture. The smallest such $G^\ast$ have @@ -332,24 +337,35 @@ 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 reachable from -some Even Level Graph via $E/O$-edge switches (``derived''), and/or -(ii) it is an intertwining tree. The conjecture holds for the class -iff at least one of (i), (ii) holds. +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 & derived only & inter.\ only & both & missing & status \\\hline -$6$ & $2$ & $0$ & $0$ & $2$ & $0$ & holds \\ -$7$ & $5$ & $0$ & $0$ & $5$ & $0$ & holds \\ -$8$ & $14$ & $0$ & $0$ & $14$ & $0$ & holds \\ -$9$ & $50$ & $0$ & $1$ & $49$ & $0$ & holds \\ -$10$ & $233$ & $0$ & $0$ & $233$ & $0$ & holds \\ -$11$ & $1249$ & $0$ & $0$ & $1249$ & $0$ & holds \\ -$12$ & $7595$ & $0$ & $1$ & $7594$ & $0$ & holds \\ +$n$ & \# 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 @@ -377,7 +393,11 @@ four, so all four are \emph{bridge-derived level graphs} derived level graphs. The witnessing orbits are small -- between a few hundred and $\sim\!1.7\times 10^5$ states -- even though other parity partitions of the same triangulations have orbits exceeding $10^6$; -finding one good partition suffices. +finding one good partition suffices. Each witness is in fact only a +\emph{handful} of bridge switches from its dual: the explicit Even Level +Graph, parity labelling, and bridge-switch sequence are recorded for all +four, with path lengths $3, 1, 2, 4$ respectively, and each step has been +verified to be a valid bridge switch. \end{itemize} Thus at $n = 21$ the disjunction is confirmed for all six critical iso classes: two are Even Level Graphs outright, and the other four are