From d9007c8697dd54f7ef6eb4b3237758c0b29bcbaa Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 19 Jun 2026 02:27:49 -0400 Subject: [PATCH] Add bridge-derived census (n=6..10) to the disjunction section Cross-tabulate bridge-derived vs intertwining-tree coverage: the bridge-derived share falls from 100% (n=6) to 62.7% (n=10), the disjunction never relies on it alone, and the "neither" column is identically zero throughout. Co-Authored-By: Claude Opus 4.8 --- .../experiments/bridge_derived_survey.py | 84 ++++++++++++++++++ papers/even_level_graph_generators/paper.aux | 16 ++-- .../paper.fdb_latexmk | 13 +-- papers/even_level_graph_generators/paper.fls | 14 ++- papers/even_level_graph_generators/paper.log | 74 +++++++-------- papers/even_level_graph_generators/paper.pdf | Bin 1384388 -> 1386477 bytes papers/even_level_graph_generators/paper.tex | 39 ++++++++ 7 files changed, 191 insertions(+), 49 deletions(-) create mode 100644 papers/even_level_graph_generators/experiments/bridge_derived_survey.py diff --git a/papers/even_level_graph_generators/experiments/bridge_derived_survey.py b/papers/even_level_graph_generators/experiments/bridge_derived_survey.py new file mode 100644 index 0000000..a8cf25e --- /dev/null +++ b/papers/even_level_graph_generators/experiments/bridge_derived_survey.py @@ -0,0 +1,84 @@ +"""Survey: for each n, how many maximal planar graphs (plane-triangulation +iso classes) are *bridge-derived* level graphs of some Even Level Graph. + +Bridge-derivedness is decided exhaustively via the backward bridge-switch +orbit (see small_n_probe.is_bridge_derived): a triangulation G is +bridge-derived iff some valid parity partition L of G admits an Even Level +Graph (parity L) in G's backward bridge-orbit. Feasible only at small n. + +Also cross-tabulates against the intertwining-tree property so the two +covering families in the disjunction conjecture can be compared. + +Usage: python3 bridge_derived_survey.py [n_max] (default 11) +""" +import sys +import os +import time +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 survey_n(n): + t0 = time.time() + tris = enumerate_all_triangulations(n) + n_bridge = 0 + n_inter = 0 + n_bridge_and_inter = 0 + n_bridge_only = 0 + n_inter_only = 0 + n_neither = 0 + for G in tris: + bd = is_bridge_derived(G) + it = is_intertwining_tree(G)[0] + if bd: + n_bridge += 1 + if it: + n_inter += 1 + if bd and it: + n_bridge_and_inter += 1 + elif bd: + n_bridge_only += 1 + elif it: + n_inter_only += 1 + else: + n_neither += 1 + return { + 'n': n, + 'total': len(tris), + 'bridge': n_bridge, + 'inter': n_inter, + 'bridge_and_inter': n_bridge_and_inter, + 'bridge_only': n_bridge_only, + 'inter_only': n_inter_only, + 'neither': n_neither, + 'elapsed': time.time() - t0, + } + + +def main(): + n_max = int(sys.argv[1]) if len(sys.argv) > 1 else 11 + rows = [] + print(f'{"n":>3} {"total":>7} {"bridge-deriv":>13} {"%":>6} ' + f'{"inter":>7} {"b&i":>6} {"b-only":>7} {"i-only":>7} ' + f'{"neither":>8} {"sec":>7}', flush=True) + for n in range(6, n_max + 1): + r = survey_n(n) + rows.append(r) + pct = 100.0 * r['bridge'] / r['total'] if r['total'] else 0.0 + print(f'{r["n"]:>3} {r["total"]:>7} {r["bridge"]:>13} {pct:>5.1f}% ' + f'{r["inter"]:>7} {r["bridge_and_inter"]:>6} ' + f'{r["bridge_only"]:>7} {r["inter_only"]:>7} ' + f'{r["neither"]:>8} {r["elapsed"]:>6.1f}', flush=True) + if r['neither']: + print(f' *** {r["neither"]} triangulation(s) at n={n} are ' + f'NEITHER bridge-derived nor intertwining trees ***', + flush=True) + return rows + + +if __name__ == '__main__': + main() diff --git a/papers/even_level_graph_generators/paper.aux b/papers/even_level_graph_generators/paper.aux index fbb687c..064f67e 100644 --- a/papers/even_level_graph_generators/paper.aux +++ b/papers/even_level_graph_generators/paper.aux @@ -44,19 +44,21 @@ \newlabel{def:intertwining-tree}{{4.6}{7}{Intertwining tree}{theorem.4.6}{}} \newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{7}{}{theorem.4.7}{}} \newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here.}}{8}{table.2}\protected@file@percent } +\newlabel{tab:bridge-census}{{2}{8}{Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here}{table.2}{}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{8}{section*.2}\protected@file@percent } \citation{holton-mckay} -\@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{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{8}{section*.3}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {3}{\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.}}{9}{table.3}\protected@file@percent } +\newlabel{tab:n21}{{3}{9}{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.3}{}} \@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.}}{9}{figure.5}\protected@file@percent } \newlabel{fig:n21-duals}{{5}{9}{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}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{10}{figure.6}\protected@file@percent } -\newlabel{fig:n24-dual}{{6}{10}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{10}{section*.3}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Beyond $n = 24$: enumeration and the next $5$-connected core}}{10}{section*.4}\protected@file@percent } -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{11}{section*.5}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{11}{figure.6}\protected@file@percent } +\newlabel{fig:n24-dual}{{6}{11}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The $25$-vertex dual $T_{25}$ of the unique $46$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only such cubic graph at $46$ vertices and the second internally $6$-connected core known. Drawn crossing-free and coloured by parity (blue even, orange odd) for its witness partition. $T_{25}$ is internally $6$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even parity subgraph.}}{12}{figure.7}\protected@file@percent } \newlabel{fig:n25-dual}{{7}{12}{The $25$-vertex dual $T_{25}$ of the unique $46$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only such cubic graph at $46$ vertices and the second internally $6$-connected core known. Drawn crossing-free and coloured by parity (blue even, orange odd) for its witness partition. $T_{25}$ is internally $6$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even parity subgraph}{figure.7}{}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{12}{section*.5}\protected@file@percent } \bibcite{holton-mckay}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{14.69437pt} diff --git a/papers/even_level_graph_generators/paper.fdb_latexmk b/papers/even_level_graph_generators/paper.fdb_latexmk index 2b74724..515a2f7 100644 --- a/papers/even_level_graph_generators/paper.fdb_latexmk +++ b/papers/even_level_graph_generators/paper.fdb_latexmk @@ -1,6 +1,5 @@ # Fdb version 3 -["pdflatex"] 1779469001 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779469003 - "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779468999 23834 39061385c4cc2522155026d2f8574bbd "" +["pdflatex"] 1781848805 "paper.tex" "paper.pdf" "paper" 1781848808 "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -20,6 +19,7 @@ "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti8.tfm" 1136768653 1504 1747189e0441d1c18f3ea56fafc1c480 "" + "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb" 1248133631 32001 6aeea3afe875097b1eb0da29acd61e28 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" @@ -33,6 +33,7 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb" 1248133631 35660 fb24af7afbadb71801619f1415838111 "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb" 1248133631 31764 459c573c03a4949a528c2cc7f557e217 "" "/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b "" "/usr/local/texlive/2022/texmf-dist/tex/generic/atbegshi/atbegshi.sty" 1575674566 24708 5584a51a7101caf7e6bbf1fc27d8f7b1 "" @@ -91,10 +92,12 @@ "fig_level_cycle.png" 1779389598 83736 ee54074ab1383a0dcc7fc20387e34bdc "" "fig_levels.png" 1779389598 88029 5564f46c0a183f3777727b651e7dc461 "" "fig_parity_subgraph.png" 1779389598 191771 f069aa94c8f49b3c7fd9c71426feff2d "" + "figures/core_n25_dual.png" 1779491939 167150 1ff2a9ce9f23b303c20e8a8910b41205 "" + "figures/fig210_dual.png" 1779469439 152438 ac3c4fe29042435cab15ea90ee80b805 "" "figures/n21_duals.png" 1779463364 667947 fd52170c20399b0c2dff901831fad5d5 "" - 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zIW{&jMKd!-I6^}{T?+~?Ol59obZ8(pIG2I-6e@ou)JsTIVHn2o?>*yDQ^#q}jMbQz zykEy_sWoL~*~RW^Q)X8M3EH$Nh@f^wwrUeZ@zJ79ZK8HABw9qdaA!lISWHm(Lt5oy_mc!1sA6p&DFA7(j!AhOKySPqHbMt*D#BXbA5~nPssaXP7uWNtecAW+&7ZbauYn5DsU2v+Q25O-W>LJcY z5+Ws@euD>e8bzwobmZ}!R^WD>HgO8S8VQU%)uVd%TcJY&=fCS(0`wQ+iEOL5cP{7w z-H&as9dv8E#C?#^qq;M?joJ+@jkZZ^b_9;XVdxR}$5KRUVe$`w1NtkgY;n zc3cznwiMW=N$ABZU8Ccc?phjavm}%n(F55Mo*CAp?aY<%hanwlvF!qSYU!z^rQm?c=L}nRZAo}rKv_DH!o|dl}PH5raFoI4QQ&DXlz zq35GlmR?zH1h0_g6|%j+dt~(wh5FbeP$^5L>;TXgJ1BYS3IC)8p7{@XM5nh3Wo~41 obaG{3Z3<;>WN%_>3UhQ}a&&ldWo8O7GdMFeG%^Y$B}Gq03TRN%DgXcg diff --git a/papers/even_level_graph_generators/paper.tex b/papers/even_level_graph_generators/paper.tex index 1af0c65..78dc694 100644 --- a/papers/even_level_graph_generators/paper.tex +++ b/papers/even_level_graph_generators/paper.tex @@ -492,6 +492,45 @@ $n = 21$ 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. +The intertwining-tree disjunct therefore carries the conjecture by itself +for all small $n$, but this leaves open how much of the load the +bridge-derived disjunct is independently able to bear. To measure that we +classified every triangulation at $6 \leq n \leq 10$ as bridge-derived or +not, deciding bridge-derivedness exhaustively: a triangulation is +bridge-derived iff some valid parity partition admits an Even Level Graph +in its backward bridge-switch orbit (a search feasible only at these +sizes). Table~\ref{tab:bridge-census} records the result. Three features +stand out. First, the bridge-derived disjunct is substantive but far from +universal on its own: its share of all triangulations falls steadily, from +all of them at $n = 6$ to under two-thirds by $n = 10$. Second, the +disjunction never relies on it in this range -- the \emph{intertwining +only} column counts triangulations covered by the tree disjunct alone, and +it grows, while no triangulation here is bridge-derived without also being +an intertwining tree. Third, and consistent with the conjecture, the +\emph{neither} column is identically zero throughout. + +\begin{table}[ht] +\centering +\begin{tabular}{cccccc} +$n$ & triangulations & bridge-derived & \% & intertwining only & neither \\\hline +$6$ & $2$ & $2$ & $100.0$ & $0$ & $0$ \\ +$7$ & $5$ & $4$ & $80.0$ & $1$ & $0$ \\ +$8$ & $14$ & $12$ & $85.7$ & $2$ & $0$ \\ +$9$ & $50$ & $36$ & $72.0$ & $14$ & $0$ \\ +$10$ & $233$ & $146$ & $62.7$ & $87$ & $0$ \\ +\end{tabular} +\caption{Bridge-derived census for $6 \leq n \leq 10$. \emph{bridge-derived} +counts plane-triangulation iso classes that are bridge-derived level graphs +of some Even Level Graph, decided by exhaustive backward bridge-switch +search over all valid parity partitions; \% is its fraction of all +triangulations. \emph{intertwining only} counts those that are intertwining +trees but not bridge-derived; \emph{neither} counts those covered by no +disjunct. Every triangulation in this range is an intertwining tree, and +every bridge-derived one is too, so bridge-derived $\subseteq$ intertwining +tree here.} +\label{tab:bridge-census} +\end{table} + \subsection*{The boundary case $n = 21$} The first triangulations that are \emph{not} intertwining trees are the