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 <noreply@anthropic.com>
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"""Survey: for each n, how many maximal planar graphs (plane-triangulation
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iso classes) are *bridge-derived* level graphs of some Even Level Graph.
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Bridge-derivedness is decided exhaustively via the backward bridge-switch
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orbit (see small_n_probe.is_bridge_derived): a triangulation G is
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bridge-derived iff some valid parity partition L of G admits an Even Level
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Graph (parity L) in G's backward bridge-orbit. Feasible only at small n.
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Also cross-tabulates against the intertwining-tree property so the two
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covering families in the disjunction conjecture can be compared.
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Usage: python3 bridge_derived_survey.py [n_max] (default 11)
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"""
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import sys
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import os
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import time
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sys.path.insert(0, '/Users/didericis/Code/math-research/papers/'
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'level_resolutions_of_maximal_planar_graphs/experiments')
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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from triangulation_gen import enumerate_all_triangulations
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from small_n_probe import is_bridge_derived
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from test_disjunction import is_intertwining_tree
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def survey_n(n):
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t0 = time.time()
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tris = enumerate_all_triangulations(n)
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n_bridge = 0
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n_inter = 0
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n_bridge_and_inter = 0
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n_bridge_only = 0
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n_inter_only = 0
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n_neither = 0
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for G in tris:
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bd = is_bridge_derived(G)
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it = is_intertwining_tree(G)[0]
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if bd:
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n_bridge += 1
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if it:
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n_inter += 1
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if bd and it:
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n_bridge_and_inter += 1
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elif bd:
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n_bridge_only += 1
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elif it:
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n_inter_only += 1
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else:
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n_neither += 1
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return {
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'n': n,
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'total': len(tris),
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'bridge': n_bridge,
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'inter': n_inter,
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'bridge_and_inter': n_bridge_and_inter,
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'bridge_only': n_bridge_only,
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'inter_only': n_inter_only,
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'neither': n_neither,
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'elapsed': time.time() - t0,
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}
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def main():
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n_max = int(sys.argv[1]) if len(sys.argv) > 1 else 11
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rows = []
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print(f'{"n":>3} {"total":>7} {"bridge-deriv":>13} {"%":>6} '
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f'{"inter":>7} {"b&i":>6} {"b-only":>7} {"i-only":>7} '
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f'{"neither":>8} {"sec":>7}', flush=True)
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for n in range(6, n_max + 1):
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r = survey_n(n)
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rows.append(r)
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pct = 100.0 * r['bridge'] / r['total'] if r['total'] else 0.0
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print(f'{r["n"]:>3} {r["total"]:>7} {r["bridge"]:>13} {pct:>5.1f}% '
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f'{r["inter"]:>7} {r["bridge_and_inter"]:>6} '
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f'{r["bridge_only"]:>7} {r["inter_only"]:>7} '
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f'{r["neither"]:>8} {r["elapsed"]:>6.1f}', flush=True)
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if r['neither']:
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print(f' *** {r["neither"]} triangulation(s) at n={n} are '
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f'NEITHER bridge-derived nor intertwining trees ***',
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flush=True)
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return rows
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if __name__ == '__main__':
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main()
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@@ -44,19 +44,21 @@
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\newlabel{def:intertwining-tree}{{4.6}{7}{Intertwining tree}{theorem.4.6}{}}
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\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{7}{}{theorem.4.7}{}}
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\newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent }
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\@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 }
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\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}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{8}{section*.2}\protected@file@percent }
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\citation{holton-mckay}
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\@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 }
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\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}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{8}{section*.3}\protected@file@percent }
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\@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 }
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\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}{}}
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\@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 }
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\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}{}}
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\@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 }
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\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}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{10}{section*.3}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Beyond $n = 24$: enumeration and the next $5$-connected core}}{10}{section*.4}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{11}{section*.5}\protected@file@percent }
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\@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 }
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\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}{}}
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\@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 }
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\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}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{Toward a characterization of bridge-derived graphs}}{12}{section*.5}\protected@file@percent }
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\bibcite{holton-mckay}{1}
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\newlabel{tocindent-1}{0pt}
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the automorphism-free count
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\OT1/cmr/m/n/10 The $\OML/cmm/m/it/10 n \OT1/cmr/m/n/10 = 23$ row re-com-putes
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Faulkner--Younger's min-i-mal-ity (no cycli-cally $5$-connected
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@@ -492,6 +492,45 @@ $n = 21$ and there are exactly $6$ of them. Below $n = 21$ every
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maximal planar graph is an intertwining tree, which is why the
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disjunction holds trivially in that range.
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The intertwining-tree disjunct therefore carries the conjecture by itself
|
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for all small $n$, but this leaves open how much of the load the
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bridge-derived disjunct is independently able to bear. To measure that we
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classified every triangulation at $6 \leq n \leq 10$ as bridge-derived or
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not, deciding bridge-derivedness exhaustively: a triangulation is
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bridge-derived iff some valid parity partition admits an Even Level Graph
|
||||
in its backward bridge-switch orbit (a search feasible only at these
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sizes). Table~\ref{tab:bridge-census} records the result. Three features
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stand out. First, the bridge-derived disjunct is substantive but far from
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universal on its own: its share of all triangulations falls steadily, from
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all of them at $n = 6$ to under two-thirds by $n = 10$. Second, the
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disjunction never relies on it in this range -- the \emph{intertwining
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only} column counts triangulations covered by the tree disjunct alone, and
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it grows, while no triangulation here is bridge-derived without also being
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an intertwining tree. Third, and consistent with the conjecture, the
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\emph{neither} column is identically zero throughout.
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\begin{table}[ht]
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\centering
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\begin{tabular}{cccccc}
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$n$ & triangulations & bridge-derived & \% & intertwining only & neither \\\hline
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$6$ & $2$ & $2$ & $100.0$ & $0$ & $0$ \\
|
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$7$ & $5$ & $4$ & $80.0$ & $1$ & $0$ \\
|
||||
$8$ & $14$ & $12$ & $85.7$ & $2$ & $0$ \\
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||||
$9$ & $50$ & $36$ & $72.0$ & $14$ & $0$ \\
|
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$10$ & $233$ & $146$ & $62.7$ & $87$ & $0$ \\
|
||||
\end{tabular}
|
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\caption{Bridge-derived census for $6 \leq n \leq 10$. \emph{bridge-derived}
|
||||
counts plane-triangulation iso classes that are bridge-derived level graphs
|
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of some Even Level Graph, decided by exhaustive backward bridge-switch
|
||||
search over all valid parity partitions; \% is its fraction of all
|
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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
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every bridge-derived one is too, so bridge-derived $\subseteq$ intertwining
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||||
tree here.}
|
||||
\label{tab:bridge-census}
|
||||
\end{table}
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||||
\subsection*{The boundary case $n = 21$}
|
||||
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||||
The first triangulations that are \emph{not} intertwining trees are the
|
||||
|
||||
Reference in New Issue
Block a user