even_level: add "Toward a characterization of bridge-derived graphs"
Record the partition sweep on the n=24 Fig 2.10 dual. New subsection + experiments/bridge_partition_sweep.py. Findings: - A bridge switch is a constrained diagonal flip; bridge-derived via L means lying in an Even-Level-Graph component of the restricted flip graph. So the question is which flip-components contain an ELG. - Identity: every 4-coloring of a triangulation has e_cross = 2n-4 (each face has one within-pair edge), so total parity-subgraph Betti = (c_A+c_B)-2; intertwining trees are the Betti-0 case. - Of T's 333 valid partitions, total Betti splits 288/42/3 over 1/2/3; min is 1 (T not intertwining). All 27 partitions found bridge-derived (depth 2-3) have the minimum Betti 1 -> necessary. - But not sufficient: only 27 of 288 Betti-1 partitions yield a witness; the rest have flip-orbits >1.5e5 with no ELG, and a 12x budget increase found none. The discriminator is flip-component structure (sharp orbit-size dichotomy), not a numerical invariant. Characterizing which Betti-minimal partitions sit in an ELG component is left open. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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"""Probe which valid parity partitions of the Fig. 2.10 dual T (24-vertex
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5-connected triangulation) make it bridge-derived, and what distinguishes
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them.
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For every valid parity partition L of T we record:
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- e_cross, e_A, e_B and the total first Betti number of the two parity
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subgraphs (which equals (c_A + c_B) - 2, since e_A + e_B = n - 2 for
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every 4-coloring of a triangulation: each triangle has exactly one
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within-colour-pair edge, 2n-4 triangles, each such edge in 2 faces);
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- whether L is the BFS-level parity of T from some source;
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- the outcome of a backward bridge-orbit search (found / capped).
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Finding: minimal total Betti (= 1 here, since T is not intertwining) is
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necessary for the bridge-derived partitions we locate, but far from
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sufficient -- the bridge-derivable partitions form a small, sharply
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separated subset distinguished by tiny flip-orbit size, not by any simple
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count. See the paper subsection "Toward a characterization of
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bridge-derived graphs".
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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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from collections import defaultdict
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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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import networkx as nx
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from sage.all import Graph # type: ignore
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from tutte_dual_treecolor import dual_triangulation
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from test_tutte_bridge import valid_parity_partitions_via_coloring, search_partition
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from test_fig210_dual_bridge import sage_to_nx
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from fast_bridge import EdgeCode
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HERE = os.path.dirname(os.path.abspath(__file__))
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def parity_stats(T, labels):
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A = [v for v in T if labels[v] == 0]
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B = [v for v in T if labels[v] == 1]
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GA, GB = T.subgraph(A), T.subgraph(B)
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cA = nx.number_connected_components(GA)
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cB = nx.number_connected_components(GB)
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bA = GA.number_of_edges() - len(A) + cA
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bB = GB.number_of_edges() - len(B) + cB
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ecross = T.number_of_edges() - GA.number_of_edges() - GB.number_of_edges()
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realiz = any(
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all((nx.single_source_shortest_path_length(T, s)[v] % 2) == labels[v] for v in T)
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or all((nx.single_source_shortest_path_length(T, s)[v] % 2) != labels[v] for v in T)
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for s in T
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)
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return dict(eA=GA.number_of_edges(), eB=GB.number_of_edges(), ecross=ecross,
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betti=bA + bB, cA=cA, cB=cB, realiz=realiz)
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def main(cap=12000, time_limit=6.0):
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g6 = open(os.path.join(HERE, 'fig210_dual.g6')).read().strip()
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T, _ = dual_triangulation(sage_to_nx(Graph(g6)))
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n = T.number_of_nodes()
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parts, _ = valid_parity_partitions_via_coloring(T)
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code = EdgeCode(T.nodes())
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code.state0 = code.state_of(T)
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print('T: n=%d, e_cross should be 2n-4=%d; %d valid partitions'
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% (n, 2 * n - 4, len(parts)), flush=True)
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by_betti = defaultdict(lambda: [0, 0]) # betti -> [found, not_found]
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ecross_vals = set()
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t0 = time.time()
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for k, labels in enumerate(parts):
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st = parity_stats(T, labels)
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ecross_vals.add(st['ecross'])
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status, sz, depth = search_partition(code, labels, n, cap, time_limit)
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slot = 0 if status == 'found' else 1
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by_betti[st['betti']][slot] += 1
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if status == 'found':
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print(' found k=%d betti=%d depth=%s orbit=%d realiz=%s'
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% (k, st['betti'], depth, sz, st['realiz']), flush=True)
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if (k + 1) % 50 == 0:
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print(' ...%d/%d (%.0fs)' % (k + 1, len(parts), time.time() - t0), flush=True)
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print('\ne_cross values over all partitions: %s (constant = identity check)'
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% sorted(ecross_vals), flush=True)
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print('betti -> [bridge-derived, not-found-at-cap=%d]:' % cap, flush=True)
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for b in sorted(by_betti):
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print(' betti=%d : %s' % (b, by_betti[b]), flush=True)
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print('(%.0fs)' % (time.time() - t0), flush=True)
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if __name__ == '__main__':
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main()
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@@ -49,15 +49,16 @@
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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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\bibcite{holton-mckay}{1}
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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 {}{}{Toward a characterization of bridge-derived graphs}}{10}{section*.4}\protected@file@percent }
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\bibcite{holton-mckay}{1}
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\newlabel{tocindent-1}{0pt}
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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 {section}{\tocsection {}{}{References}}{10}{section*.4}\protected@file@percent }
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@@ -622,6 +622,58 @@ in particular no odd cycle, is created.}
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\label{fig:n24-dual}
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\end{figure}
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\subsection*{Toward a characterization of bridge-derived graphs}
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A bridge switch is a diagonal flip of the quadrilateral around a level
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edge, constrained so the flipped-in edge enters no parity subgraph as a
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cycle edge. Fixing a valid parity partition $L$, the bridge switches
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therefore act on the triangulations whose two $L$-parity subgraphs are
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both bipartite, and $G$ is a bridge-derived level graph via $L$ exactly
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when $G$ lies in the same connected component as an Even Level Graph in
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this restricted flip graph. So the question ``which triangulations are
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bridge-derived?'' is really ``which flip-components contain an Even Level
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Graph?'', quantified over $L$. We probed this on the $24$-vertex dual $T$
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of Figure~\ref{fig:n24-dual}, sweeping all of its valid parity partitions;
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the experiment is recorded in \texttt{experiments/bridge\_partition\_sweep.py}.
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One coordinate organises the picture. For any proper $4$-coloring of a
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triangulation, group the colours as $\{1,2\}\mid\{3,4\}$; then each
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triangular face has exactly one within-pair edge, and since there are
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$2n-4$ faces and each edge lies on two of them, the parity subgraphs carry
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exactly $n-2$ edges between them. Hence $e_{\mathrm{cross}} = 2n-4$ for
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\emph{every} valid partition (confirmed across all partitions of $T$), and
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the total first Betti number of the two parity subgraphs equals
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$(c_A + c_B) - 2$, where $c_A, c_B$ count their connected components. The
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intertwining-tree case is precisely total Betti $0$ -- both parts trees,
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$c_A = c_B = 1$ -- so for a triangulation that is not an intertwining tree
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the total Betti is at least $1$.
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The $333$ valid partitions of $T$ have total Betti $1$, $2$, $3$ for
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$288$, $42$, $3$ of them respectively; the minimum is $1$, consistent with
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$T$ not being an intertwining tree. A backward bridge-orbit search locates
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Even Level Graph witnesses (at depth $2$--$3$) for $27$ partitions, and
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\emph{every} one of them has the minimum total Betti $1$ -- one parity
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class a tree, the other a tree plus a single even cycle. Minimal total
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||||
Betti is thus a necessary feature of the bridge-derived partitions we
|
||||
find.
|
||||
|
||||
It is not sufficient, and the way it fails is informative. Of the $288$
|
||||
Betti-$1$ partitions only those $27$ yield a witness; the rest exhibit
|
||||
bridge-orbits exceeding $1.5\times 10^5$ states with no Even Level Graph,
|
||||
and increasing the search budget twelvefold produced no further witnesses.
|
||||
The bridge-derivable partitions are separated from the others not by any
|
||||
of the simple invariants we measured -- total Betti, component counts,
|
||||
class sizes, or BFS-level realizability (uniformly false here) -- but by a
|
||||
sharp dichotomy in flip-orbit size: a tiny component containing an Even
|
||||
Level Graph versus a vast one that appears not to. (We cannot yet certify
|
||||
the latter, as no large orbit was exhausted; but a twelvefold budget
|
||||
increase yielding nothing makes mere depth an unlikely explanation.) The
|
||||
evidence therefore points away from a numerical characterization and
|
||||
toward the component structure of the restricted flip graph: minimal total
|
||||
Betti is a clean necessary condition, but characterizing \emph{which}
|
||||
Betti-minimal partitions lie in an Even-Level-Graph component remains open
|
||||
and is, on this evidence, the crux of deciding bridge-derivability.
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{holton-mckay}
|
||||
|
||||
Reference in New Issue
Block a user