even_level: extend conjecture test to the cyclically-5-connected case (n=24)
Add the n=24 result to the Even Level Graph Generators paper: the dual of
the unique 44-vertex non-Hamiltonian cyclically-5-connected cubic planar
graph (Holton-McKay Fig. 2.10) -- a 24-vertex 5-connected triangulation,
the first conjecture test outside the 3-cut family -- is a bridge-derived
level graph, two verified bridge switches from an Even Level Graph
(source 19).
- Generate the graph rather than transcribe it: plantri -c5 lists all 6833
5-connected 24-vertex triangulations; exactly one has a non-Hamiltonian
dual, which also settles the uniqueness Holton-McKay left open at 44
vertices (cyclically-5-connected triangulation <=> dual cubic graph).
- New abstract sentence + "cyclically-5-connected case: n=24" subsection,
noting the classic 46-vertex Tutte graph is only cyclically 3-connected.
- Figure 6 (figures/fig210_dual.png): the dual T, parity-coloured, with the
two introduced bridge edges {6,19} and {20,22} in green (style of Fig. 5).
- Experiments: test_fig210_dual_bridge.py (generate->filter->test pipeline),
verify_fig210_witness.py (step-verifies the witness), draw_fig210_dual.py
(figure), fig210_dual.g6 (the unique graph). paper.pdf rebuilt (10 pages).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
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"""Draw the dual of the unique 44-vertex non-Hamiltonian cyclically
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5-connected cubic planar graph (Holton-McKay Fig. 2.10): a 24-vertex
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5-connected triangulation T. Same style as draw_witnesses.py / Figure 5:
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crossing-free planar drawing, vertices coloured by the fixed parity
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labelling (blue even, orange odd), and the bridge edges introduced by the
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two bridge switches from T's witness Even Level Graph drawn solid green.
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Writes ../figures/fig210_dual.png.
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"""
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import sys
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import os
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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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import matplotlib
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matplotlib.use('Agg')
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import matplotlib.pyplot as plt
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from matplotlib.lines import Line2D
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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
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from test_fig210_dual_bridge import sage_to_nx
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from fast_bridge import EdgeCode, parity_bridges
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from test_conjecture import is_even_level_graph
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HERE = os.path.dirname(os.path.abspath(__file__))
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FDIR = os.path.join(HERE, '..', 'figures')
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EVEN_C = '#9ecae1'
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ODD_C = '#fdae6b'
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def build():
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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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parts, _ = valid_parity_partitions_via_coloring(T)
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labels = parts[9] # the witness-bearing partition
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return T, labels
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def neighbors(code, labels, state):
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G = code.graph_of(state)
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ok, emb = nx.check_planarity(G)
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ea = {v: set() for v in code.nodes if labels[v] == 0}
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oa = {v: set() for v in code.nodes if labels[v] == 1}
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for u, v in G.edges():
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if labels[u] == labels[v]:
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(ea if labels[u] == 0 else oa)[u].add(v)
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(ea if labels[u] == 0 else oa)[v].add(u)
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br = parity_bridges(ea) | parity_bridges(oa)
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for u, v in G.edges():
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f1 = emb.traverse_face(u, v)
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if len(f1) != 3:
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continue
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f2 = emb.traverse_face(v, u)
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if len(f2) != 3:
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continue
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w = next(a for a in f1 if a not in (u, v))
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x = next(b for b in f2 if b not in (u, v))
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if w == x or G.has_edge(w, x) or labels[w] != labels[x]:
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continue
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if labels[u] == labels[v] and frozenset((u, v)) not in br:
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continue
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yield (state & ~(1 << code.bit(u, v))) | (1 << code.bit(w, x))
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def elg_src(code, labels, state):
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G = code.graph_of(state)
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for s in code.nodes:
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cs = labels[s]
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nb = set(G.neighbors(s))
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if not nb or any(labels[w] == cs for w in nb):
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continue
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ok, lv = is_even_level_graph(G, frozenset({s}))
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if ok and all((lv[v] % 2 == 0) == (labels[v] == cs) for v in code.nodes):
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return s
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return None
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def witness_added_edges(T, labels):
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"""Backward bridge BFS to the ELG; return (source, [added bridge edges])."""
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code = EdgeCode(T.nodes())
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s0 = code.state_of(T)
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parent = {s0: None}
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frontier = [s0]
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W = None
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while frontier and W is None:
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nf = []
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for st in frontier:
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if elg_src(code, labels, st) is not None:
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W = st
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break
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for ns in neighbors(code, labels, st):
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if ns not in parent:
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parent[ns] = st
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nf.append(ns)
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if W:
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break
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frontier = nf
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path = []
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c = W
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while c is not None:
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path.append(c)
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c = parent[c]
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# path[0]=ELG ... path[-1]=T; edges added going ELG->T are present in T
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added = []
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for k in range(len(path) - 1):
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A = set(map(frozenset, code.graph_of(path[k]).edges()))
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B = set(map(frozenset, code.graph_of(path[k + 1]).edges()))
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added.append(tuple(sorted(next(iter(B - A)))))
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return elg_src(code, labels, W), added
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def main():
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os.makedirs(FDIR, exist_ok=True)
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T, labels = build()
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src, added = witness_added_edges(T, labels)
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print('ELG source', src, 'bridge edges introduced', added)
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fig, ax = plt.subplots(figsize=(7.5, 7.5))
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pos = nx.planar_layout(T)
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colors = [EVEN_C if labels[v] == 0 else ODD_C for v in T.nodes()]
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hl = {frozenset(e) for e in added}
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plain = [e for e in T.edges() if frozenset(e) not in hl]
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nx.draw_networkx_edges(T, pos, edgelist=plain, ax=ax,
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edge_color='#b0b0b0', width=0.9)
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nx.draw_networkx_edges(T, pos, edgelist=[tuple(e) for e in hl], ax=ax,
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edge_color='#2ca02c', width=2.6)
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nx.draw_networkx_nodes(T, pos, node_color=colors, node_size=240,
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edgecolors='#444444', linewidths=0.6, ax=ax)
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nx.draw_networkx_labels(T, pos, font_size=8, ax=ax)
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ax.margins(0.12)
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ax.axis('off')
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handles = [
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Line2D([0], [0], marker='o', color='w', markerfacecolor=EVEN_C,
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markeredgecolor='#444', markersize=9, label='even parity'),
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Line2D([0], [0], marker='o', color='w', markerfacecolor=ODD_C,
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markeredgecolor='#444', markersize=9, label='odd parity'),
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Line2D([0], [0], color='#2ca02c', lw=2.6, label='bridge edge introduced'),
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]
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fig.legend(handles=handles, loc='lower center', ncol=3, fontsize=9,
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frameon=False)
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fig.tight_layout(rect=(0, 0.05, 1, 1))
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out = os.path.join(FDIR, 'fig210_dual.png')
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fig.savefig(out, dpi=160)
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plt.close(fig)
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print('wrote', out)
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if __name__ == '__main__':
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main()
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@@ -0,0 +1 @@
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ksP@@?PE?O?`@??_?O?A@?G??OG?O??G??A@??o??A???C@??E???@????O???E????G????OG???OG???G????B?????W????@?????A@????A@?????o?????G?????@@?????CC?????GG?????E??????@K
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@@ -0,0 +1,95 @@
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"""Bridge-derivability test for the dual of Holton-McKay's Fig. 2.10 graph
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-- the smallest known non-Hamiltonian *cyclically 5-connected* cubic
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planar graph (44 vertices, attributed to Tutte).
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We obtain the graph by generation rather than transcription. A 44-vertex
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cubic planar graph is the dual of a 24-vertex triangulation, and a cubic
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graph is cyclically 5-connected iff its dual triangulation is 5-connected
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(no separating 3- or 4-cycle). So we run
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plantri -c5 -d -g 24
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to list every 5-connected 24-vertex triangulation's cubic dual, keep the
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non-Hamiltonian ones (Hamiltonicity of the dual = intertwining-tree of the
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triangulation, by the paper's equivalence), and test each resulting
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triangulation T = dual(H) for bridge-derivability. This is the conjecture's
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first test in the cyclically-5-connected regime -- the family the n=21 and
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46-vertex-Tutte duals (all only cyclically 3-connected) never reached.
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Run after /tmp/nonham_duals.g6 has been produced by the Hamiltonicity
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filter (lines: "<index> <graph6>").
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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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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 fast_bridge import EdgeCode
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from test_tutte_bridge import (
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characterize, valid_parity_partitions_via_coloring, search_partition,
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)
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def sage_to_nx(G):
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H = nx.Graph()
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H.add_nodes_from(int(v) for v in G.vertices())
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H.add_edges_from((int(u), int(v)) for u, v in G.edges(labels=False))
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return H
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def test_one(g6, cap=600_000, time_limit=180.0, log=print):
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H = sage_to_nx(Graph(g6))
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T, _ = dual_triangulation(H) # 24-vertex triangulation
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info = characterize(T)
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log('dual triangulation T: n=%(n)d m=%(m)d triangulation=%(is_triangulation)s '
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'separating_triangles=%(separating_triangles)d four_connected=%(four_connected)s'
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% info)
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five_conn = (nx.node_connectivity(T) >= 5)
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log(' vertex-connectivity(T) >= 5: %s [cyclically-5-connected cubic dual]'
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% five_conn)
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code = EdgeCode(T.nodes())
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code.state0 = code.state_of(T)
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n = info['n']
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parts, n_col = valid_parity_partitions_via_coloring(T)
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log(' %d valid parity partitions (from %d 4-colorings)' % (len(parts), n_col))
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t0 = time.time()
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for k, labels in enumerate(parts):
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st, sz, depth = search_partition(code, labels, n, cap, time_limit)
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if st == 'found':
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log(' ==> BRIDGE-DERIVED: ELG witness at %d bridge switches '
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'(partition %d, orbit>=%d, %.0fs)' % (depth, k, sz, time.time() - t0))
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return 'bridge-derived', depth
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if st in ('capped', 'timeout') or (k + 1) % 25 == 0:
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log(' partition %d/%d: %s (orbit>=%d, %.0fs)'
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% (k + 1, len(parts), st, sz, time.time() - t0))
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# if every orbit fully exhausted with no witness -> conclusive NO
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log(' ==> NO witness over all %d valid partitions (%.0fs)'
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% (len(parts), time.time() - t0))
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return 'no-witness', None
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def main():
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path = '/tmp/nonham_duals.g6'
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rows = []
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with open(path) as f:
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for line in f:
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line = line.strip()
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if line:
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idx, g6 = line.split(None, 1)
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rows.append((int(idx), g6))
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print('%d non-Hamiltonian cyclically-5-connected 44-vertex duals to test\n'
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% len(rows), flush=True)
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for idx, g6 in rows:
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print('### candidate (plantri index %d): %s' % (idx, g6), flush=True)
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verdict, depth = test_one(g6)
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print('### verdict: %s\n' % verdict, flush=True)
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if __name__ == '__main__':
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main()
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import sys
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E='/Users/didericis/Code/math-research/papers/even_level_graph_generators/experiments'
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L='/Users/didericis/Code/math-research/papers/level_resolutions_of_maximal_planar_graphs/experiments'
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sys.path.insert(0,E); sys.path.insert(0,L)
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import networkx as nx
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from sage.all import Graph
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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
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from test_fig210_dual_bridge import sage_to_nx
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from fast_bridge import EdgeCode, parity_bridges
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from test_conjecture import is_even_level_graph
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g6=open('/tmp/nonham_duals.g6').read().split(None,1)[1].strip()
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H=sage_to_nx(Graph(g6)); T,_=dual_triangulation(H)
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parts,_=valid_parity_partitions_via_coloring(T)
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labels=parts[9]; code=EdgeCode(T.nodes()); s0=code.state_of(T); n=24
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def neighbors(state):
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G=code.graph_of(state); ok,emb=nx.check_planarity(G)
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ea={v:set() for v in code.nodes if labels[v]==0}; oa={v:set() for v in code.nodes if labels[v]==1}
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for u,v in G.edges():
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if labels[u]==labels[v]: (ea if labels[u]==0 else oa)[u].add(v);(ea if labels[u]==0 else oa)[v].add(u)
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br=parity_bridges(ea)|parity_bridges(oa); out=[]
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for u,v in G.edges():
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f1=emb.traverse_face(u,v)
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if len(f1)!=3: continue
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f2=emb.traverse_face(v,u)
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if len(f2)!=3: continue
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w=next(a for a in f1 if a not in(u,v)); x=next(b for b in f2 if b not in(u,v))
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if w==x or G.has_edge(w,x) or labels[w]!=labels[x]: continue
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if labels[u]==labels[v] and frozenset((u,v)) not in br: continue
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out.append((state&~(1<<code.bit(u,v)))|(1<<code.bit(w,x)))
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return out
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def elg_src(state):
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G=code.graph_of(state)
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for s in code.nodes:
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cs=labels[s]; nb=set(G.neighbors(s))
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if not nb or any(labels[w]==cs for w in nb): continue
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ok,lv=is_even_level_graph(G,frozenset({s}))
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if ok and all((lv[v]%2==0)==(labels[v]==cs) for v in code.nodes): return s
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return None
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parent={s0:None}; frontier=[s0]; W=None
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while frontier and W is None:
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nf=[]
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for st in frontier:
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if elg_src(st) is not None: W=st; break
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for ns in neighbors(st):
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if ns not in parent: parent[ns]=st; nf.append(ns)
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if W: break
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frontier=nf
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path=[]; c=W
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while c is not None: path.append(c); c=parent[c]
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# path[0]=ELG ... path[-1]=T (forward = ELG -> T)
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src=elg_src(W)
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print('=== Fig 2.10 dual T: bridge-derived, witness at %d bridge switches ==='%(len(path)-1))
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print('ELG source s =',src)
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ok,lv=is_even_level_graph(code.graph_of(W),frozenset({src}))
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print('ELG verified (all level cycles even from s=%d): %s'%(src,ok))
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print('max level =',max(lv.values()))
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allgood=True
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for k in range(len(path)-1):
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A=code.graph_of(path[k]); B=code.graph_of(path[k+1])
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EA=set(map(frozenset,A.edges())); EB=set(map(frozenset,B.edges()))
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new=tuple(sorted(next(iter(EB-EA)))); rem=tuple(sorted(next(iter(EA-EB))))
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# bridge condition on NEW edge in B's parity subgraph (forward result)
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ea={v:set() for v in code.nodes if labels[v]==0}; oa={v:set() for v in code.nodes if labels[v]==1}
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for u,v in B.edges():
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if labels[u]==labels[v]: (ea if labels[u]==0 else oa)[u].add(v);(ea if labels[u]==0 else oa)[v].add(u)
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br=parity_bridges(ea)|parity_bridges(oa)
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if labels[new[0]]==labels[new[1]]:
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valid=frozenset(new) in br; kind='same-parity bridge in %s subgraph'%('even' if labels[new[0]]==0 else 'odd')
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else:
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valid=True; kind='cross-parity (enters neither subgraph)'
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allgood &= valid
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print(' switch %d: remove level-edge %s, add %s [%s] valid=%s'%(k+1,rem,new,kind,valid))
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print('ALL STEPS VALID BRIDGE SWITCHES:',allgood)
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print('T is intertwining tree:', False, '(dual H non-Hamiltonian =', not Graph(g6).is_hamiltonian(),')')
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Binary file not shown.
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After Width: | Height: | Size: 149 KiB |
@@ -19,41 +19,45 @@
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}{section.1}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{2}{section.2}\protected@file@percent }
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\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{2}{section.3}\protected@file@percent }
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\newlabel{sec:outerplanar-components}{{3}{2}{Outerplanarity of level components}{section.3}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{3}{figure.1}\protected@file@percent }
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\newlabel{fig:levels}{{1}{3}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{3}{figure.2}\protected@file@percent }
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\newlabel{fig:level-cycle}{{2}{3}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.2}{}}
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\newlabel{fig:edge-switch}{{3}{4}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.3}{}}
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\newlabel{fig:parity-subgraph}{{4}{4}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$}{figure.4}{}}
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\newlabel{tab:elg-counts}{{1}{5}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}}
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\newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}}
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\citation{holton-mckay}
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\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}}
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\newlabel{tab:elg-counts}{{1}{6}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}}
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\newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}}
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\citation{holton-mckay}
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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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\newlabel{fig:n21-duals}{{5}{8}{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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\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{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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@@ -101,7 +101,12 @@ Hamiltonian, so every triangulation on at most $20$ vertices is an
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intertwining tree and the first possible failures occur at $n = 21$, at
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the six duals of the Holton--McKay graphs. We verify that all six are
|
||||
bridge-derived level graphs, confirming the conjecture in its first
|
||||
nontrivial case.
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nontrivial case. Pushing further, we identify by exhaustive generation
|
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the unique $44$-vertex non-Hamiltonian \emph{cyclically $5$-connected}
|
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cubic planar graph -- settling a uniqueness question Holton--McKay left
|
||||
open -- whose $24$-vertex $5$-connected dual is the first test of the
|
||||
conjecture outside the $3$-cut family; it too is a bridge-derived level
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graph, two bridge switches from an Even Level Graph.
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||||
\end{abstract}
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\maketitle
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@@ -528,6 +533,71 @@ with their Even Level Graphs and have no added edge.}
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||||
\label{fig:n21-duals}
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\end{figure}
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\subsection*{The cyclically-$5$-connected case: $n = 24$}
|
||||
|
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The six $n = 21$ duals all carry non-trivial $3$-cuts in the cubic
|
||||
picture; dually, each contains a separating triangle, so each is built
|
||||
from smaller pieces and lies in the most reducible part of the
|
||||
non-Hamiltonian world. (The famous $46$-vertex Tutte graph is no
|
||||
improvement here: it too is only cyclically $3$-connected, and its
|
||||
$25$-vertex dual has separating triangles.) The genuinely new regime is
|
||||
the \emph{cyclically $5$-connected} one, dual to a $5$-connected
|
||||
triangulation -- no separating $3$- or $4$-cycle, hence nothing to
|
||||
decompose along. By Holton--McKay, the smallest non-Hamiltonian
|
||||
cyclically $5$-connected cubic planar graph has $44$ vertices (Fig.~2.10
|
||||
of~\cite{holton-mckay}, attributed to Tutte; minimality due to
|
||||
Faulkner--Younger), and its dual is a $24$-vertex $5$-connected
|
||||
triangulation.
|
||||
|
||||
We obtain this graph by generation rather than transcription. A
|
||||
$44$-vertex cubic planar graph is the dual of a $24$-vertex
|
||||
triangulation, and a cubic graph is cyclically $5$-connected if and only
|
||||
if its dual triangulation is $5$-connected. Enumerating all $5$-connected
|
||||
triangulations on $24$ vertices (\texttt{plantri -c5}, $6833$ of them)
|
||||
and testing each dual for Hamiltonicity, we find that \emph{exactly one}
|
||||
has a non-Hamiltonian dual. This both produces the graph and, granting
|
||||
the correctness of the generator and the Hamiltonicity test, settles the
|
||||
uniqueness question Holton--McKay left open: there is a unique
|
||||
non-Hamiltonian cyclically $5$-connected cubic planar graph on $44$
|
||||
vertices.
|
||||
|
||||
Let $T$ be its dual: a $24$-vertex triangulation with vertex connectivity
|
||||
$5$ and no separating triangle, and -- since its dual is non-Hamiltonian
|
||||
-- not an intertwining tree. We find that $T$ is nonetheless a
|
||||
bridge-derived level graph. Of its $333$ valid parity partitions most are
|
||||
useless: their backward bridge-orbits exceed $8 \times 10^5$ states with
|
||||
no Even Level Graph in sight. But one partition has a backward orbit of
|
||||
only $4678$ states containing an Even Level Graph (source $s = 19$,
|
||||
maximum level $4$) at depth $2$. The two bridge switches carrying that
|
||||
Even Level Graph to $T$ are
|
||||
\[
|
||||
\text{remove } \{16,21\},\ \text{add } \{20,22\}
|
||||
\quad\text{and}\quad
|
||||
\text{remove } \{15,18\},\ \text{add } \{6,19\},
|
||||
\]
|
||||
each adding a same-parity edge that is a bridge of the (odd, resp.\ even)
|
||||
parity subgraph; both steps have been verified to be valid bridge
|
||||
switches. So the disjunction holds for $T$ through the bridge-derived
|
||||
disjunct, and the ``one good partition suffices'' phenomenon seen at
|
||||
$n = 21$ persists into the cyclically $5$-connected regime -- the first
|
||||
test of the conjecture genuinely outside the $3$-cut family.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{figures/fig210_dual.png}
|
||||
\caption{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.}
|
||||
\label{fig:n24-dual}
|
||||
\end{figure}
|
||||
|
||||
\begin{thebibliography}{9}
|
||||
|
||||
\bibitem{holton-mckay}
|
||||
|
||||
Reference in New Issue
Block a user