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>

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()

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 }
+\@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}{{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}{}}
+\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 }
 \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 }
+\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{10}{section*.3}\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 {}{}{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}

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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 
 Faulkner--Younger's min-i-mal-ity (no cycli-cally $5$-connected
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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