From 005cbba53b71d112cb51a4c48b6820900ed8e9f1 Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 22 May 2026 11:27:55 -0400 Subject: [PATCH] Remove the Empirical status subsection (small-n table) Drop the n<=9 bridge-derived classification table and its surrounding discussion; the n=21 boundary case now follows directly from the trivial-below-21 observation. Co-Authored-By: Claude Opus 4.7 --- papers/even_level_graph_generators/paper.pdf | Bin 1079927 -> 1076840 bytes papers/even_level_graph_generators/paper.tex | 32 ------------------- 2 files changed, 32 deletions(-) diff --git a/papers/even_level_graph_generators/paper.pdf b/papers/even_level_graph_generators/paper.pdf index 2c98eca876ebc3d19780a273b4e7ef524f0109ea..f790085c4965f84a1b144994f697f2a2b0bc3ee4 100644 GIT binary patch delta 73400 zcmX`RQ*bUy7p@!In6Yi!wr$&XzSy>HJ2SSe8QZqC*T3uRbMaL5VE0W|b-y)UpM=h~ zgpkP;MaAiu=-FV%=9fp-U|3ihk^ez}FmtB&N`o>cV?l}mGc_APCVv4l{XY*^x*-S% zV!I6(uqQYO8)qUiYWo{Ba1CU7k2EM#It2&_S2}?SC@cs&TRRR4@a8Qr+yAX>|1Z)7 z4$S;N!|MObU=P=V0%_NR0yPl^VP|juFTxH3!p_zHq6*p%pC0Iegql7t4N8!HW(+C} z!od+vYXbToVC}Rf;O3>m<|D6fG9m@|~0S1_*-6|YB1`(5si;$7f!Ppvx zmluXX*390*)sm2fla)E$2?T@^(4#Gryw!%>`>An7mqQIN%L^5x{!TbzQ;9MnKcO=* z@DiA{e=~oOfRj|5Dh=S@K7$}=Uen`pKNdv-y0>%%e7glR)c1aT_-N^BDjjX;>FOC! 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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