even_level: extend to n=25 -- second internally-6-connected core, also bridge-derived
Enumerate non-Hamiltonian cyclically-5-connected cubic planar graphs by
running plantri -c5 -d for n in {23,25,26} (n=24 already in the previous
commit) and filtering for non-Hamiltonian dual:
n=23 -> 0 of 1970 (recomputes Faulkner-Younger minimality)
n=24 -> 1 of 6833 (the Tutte/Fig 2.10 graph)
n=25 -> 1 of 23384 (new; unique 46-vertex one)
n=26 -> 0 of 82625
Both T (n=24) and T_25 (n=25) verified internally 6-connected by exhaustive
5-cut scan: every 5-cut is the neighborhood of a degree-5 vertex. This is
the strongest connectivity a planar triangulation can have and the level
at which Birkhoff-style reductions terminate, so both are genuinely
irreducible bases of any decomposition argument.
T_25 is also bridge-derived: witness Even Level Graph from source 24
(max level 4) at depth 2, orbit only 3114 states. Forward switches:
remove {21,23} add {22,24}; remove {3,5} add {1,6}. Both adds are bridges
of the even parity subgraph. Same witness signature as T (minimum total
Betti, tiny orbit, depth 2).
New subsection "Beyond n=24: enumeration and the next 5-connected core",
abstract extended, new Figure 7 (core_n25_dual.png). Reproducibility
scripts: draw_core_witness.py and verify_core_witness.py (both
parametrized so they work on any 5-conn non-Ham-dual core's g6).
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
@@ -0,0 +1 @@
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msP@@?PE?O?`@??_?O?A@?G??OG?O??G??A@??o??A???C@??C???A????_???C????o????_????_G???OC???E?????_????A?_???K?????K?????C?????@_G????B??????CC?????G??????GG?????CC?????@@??????BG
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@@ -0,0 +1,171 @@
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"""Draw a 5-connected non-Hamiltonian-dual core (= the dual of a non-Hamiltonian
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cyclically 5-connected cubic planar graph) as a parity-coloured planar graph
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with the bridge edges introduced by its witness Even Level Graph highlighted.
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Style matches Fig. 6 (the n=24 core).
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The script picks the first valid parity partition of minimal total Betti for
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which a backward bridge-orbit search returns an Even Level Graph, and prints
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the witness data (source, added bridge edges).
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Usage:
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sage -python draw_core_witness.py <input_g6_file> <output_png_path>
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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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EVEN_C = '#9ecae1'
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ODD_C = '#fdae6b'
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def betti(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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return (GA.number_of_edges() - len(A) + nx.number_connected_components(GA)) + \
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(GB.number_of_edges() - len(B) + nx.number_connected_components(GB))
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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 find_witness(T, labels, cap):
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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 and len(parent) < cap:
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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 len(parent) >= cap:
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break
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if W or len(parent) >= cap:
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break
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if W:
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break
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frontier = nf
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if W is None:
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return None
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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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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, len(path) - 1
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def main():
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if len(sys.argv) < 3:
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print('usage: draw_core_witness.py <input_g6_file> <output_png_path>')
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sys.exit(1)
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in_g6_path, out_png = sys.argv[1], sys.argv[2]
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g6 = open(in_g6_path).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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order = sorted(range(len(parts)), key=lambda k: betti(T, parts[k]))
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chosen = None
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for k in order[:200]:
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r = find_witness(T, parts[k], cap=40000)
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if r is not None:
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chosen = (k, parts[k], r)
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break
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if chosen is None:
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print('no witness found in scanned partitions; aborting')
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sys.exit(2)
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k, labels, (src, added, depth) = chosen
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print('partition=%d source=%d depth=%d added=%s' % (k, src, depth, added))
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fig, ax = plt.subplots(figsize=(8, 8))
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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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os.makedirs(os.path.dirname(out_png), exist_ok=True)
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fig.savefig(out_png, dpi=160)
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plt.close(fig)
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print('wrote', out_png)
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if __name__ == '__main__':
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main()
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@@ -0,0 +1,101 @@
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"""Step-verify the bridge-derivability witness for a 5-connected
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non-Hamiltonian-dual core. Same outputs as verify_fig210_witness.py
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(parity partition, ELG source/max level, both removed and added edges per
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forward switch, bridge-condition validity check) but parametrized so it
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works on any core's graph6.
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Usage:
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sage -python verify_core_witness.py <input_g6_file>
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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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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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from draw_core_witness import betti, neighbors, elg_src, find_witness
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def main():
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if len(sys.argv) < 2:
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print('usage: verify_core_witness.py <input_g6_file>')
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sys.exit(1)
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g6 = open(sys.argv[1]).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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order = sorted(range(len(parts)), key=lambda k: betti(T, parts[k]))
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chosen = None
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for k in order[:200]:
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r = find_witness(T, parts[k], cap=40000)
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if r is not None:
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chosen = (k, parts[k], r)
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break
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assert chosen is not None, 'no witness found'
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k, labels, (src, _, _) = chosen
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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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ok, lv = is_even_level_graph(code.graph_of(W), frozenset({src}))
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print('partition %d, source %d, max level %d, depth %d, ELG verified %s'
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% (k, src, max(lv.values()), len(path) - 1, ok))
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print('parity even =', sorted(v for v in T if labels[v] == 0))
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print('parity odd =', sorted(v for v in T if labels[v] == 1))
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all_valid = True
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for i in range(len(path) - 1):
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A = set(map(frozenset, code.graph_of(path[i]).edges()))
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B = set(map(frozenset, code.graph_of(path[i + 1]).edges()))
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new = tuple(sorted(next(iter(B - A))))
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rem = tuple(sorted(next(iter(A - B))))
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Gp = code.graph_of(path[i + 1])
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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 Gp.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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if labels[new[0]] == labels[new[1]]:
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valid = frozenset(new) in br
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kind = 'same-parity bridge in %s subgraph' % (
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'even' if labels[new[0]] == 0 else 'odd')
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else:
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valid = True
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kind = 'cross-parity (enters neither subgraph)'
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all_valid &= valid
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print(' switch %d: remove %s, add %s [%s] valid=%s'
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% (i + 1, rem, new, kind, valid))
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print('ALL STEPS VALID BRIDGE SWITCHES:', all_valid)
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if __name__ == '__main__':
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main()
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|
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@@ -17,19 +17,19 @@
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\providecommand\HyField@AuxAddToFields[1]{}
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\providecommand\HyField@AuxAddToCoFields[2]{}
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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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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Definitions}}{3}{section.2}\protected@file@percent }
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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{def:edge-switch}{{2.4}{4}{Edge switch}{theorem.2.4}{}}
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\newlabel{def:edge-switch}{{2.4}{3}{Edge switch}{theorem.2.4}{}}
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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 $1$.}}{4}{figure.2}\protected@file@percent }
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\newlabel{fig:level-cycle}{{2}{4}{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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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \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 $G$.}}{4}{figure.3}\protected@file@percent }
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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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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (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 $G'$.}}{4}{figure.4}\protected@file@percent }
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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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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{5}{section.3}\protected@file@percent }
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\newlabel{sec:outerplanar-components}{{3}{5}{Outerplanarity of level components}{section.3}{}}
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\newlabel{thm:outerplanar-component}{{3.1}{5}{}{theorem.3.1}{}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{4}{section.3}\protected@file@percent }
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\newlabel{sec:outerplanar-components}{{3}{4}{Outerplanarity of level components}{section.3}{}}
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\newlabel{thm:outerplanar-component}{{3.1}{4}{}{theorem.3.1}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (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 $G'$.}}{5}{figure.4}\protected@file@percent }
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\newlabel{fig:parity-subgraph}{{4}{5}{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{def:even-level-graph}{{4.1}{5}{Even Level Graph}{theorem.4.1}{}}
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@@ -39,8 +39,8 @@
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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:derived-level-graph}{{4.3}{6}{Derived level graph}{theorem.4.3}{}}
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\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}}
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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{def:bridge-derived-level-graph}{{4.5}{7}{Bridge-derived level graph}{theorem.4.5}{}}
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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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||||
\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{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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@@ -112,7 +112,15 @@ the unique $44$-vertex non-Hamiltonian \emph{cyclically $5$-connected}
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cubic planar graph -- settling a uniqueness question Holton--McKay left
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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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graph, two bridge switches from an Even Level Graph. Iterating the same
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generation procedure up to $n = 26$ produces, at $n = 25$, a second
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non-Hamiltonian cyclically-$5$-connected cubic planar graph (likewise
|
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unique at its size); its dual is verified \emph{internally
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$6$-connected} -- the strongest connectivity possible for a planar
|
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triangulation, and the level at which Birkhoff-style reductions
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terminate -- and likewise bridge-derived, again at depth $2$. Both known
|
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internally-$6$-connected non-Hamiltonian-dual cores thus satisfy the
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conjecture, with identical witness signature.
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\end{abstract}
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\maketitle
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@@ -586,8 +594,8 @@ non-Hamiltonian cyclically $5$-connected cubic planar graph on $44$
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vertices.
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Let $T$ be its dual: a $24$-vertex triangulation with vertex connectivity
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$5$ and no separating triangle, and -- since its dual is non-Hamiltonian
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-- not an intertwining tree. We find that $T$ is nonetheless a
|
||||
$5$ (in fact internally $6$-connected, verified in the next subsection),
|
||||
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
|
||||
@@ -622,6 +630,80 @@ in particular no odd cycle, is created.}
|
||||
\label{fig:n24-dual}
|
||||
\end{figure}
|
||||
|
||||
\subsection*{Beyond $n = 24$: enumeration and the next $5$-connected core}
|
||||
|
||||
The graph $T$ of the previous subsection is one core; iterating the same
|
||||
generation procedure produces the rest. Below, the third column counts
|
||||
$5$-connected triangulations on $n$ vertices (\texttt{plantri -c5} $n$),
|
||||
and the fourth filters them by Hamiltonicity of the cubic dual:
|
||||
\begin{center}
|
||||
\begin{tabular}{cccc}
|
||||
$n$ & dual $|V|$ & $5$-connected triangulations & non-Hamiltonian dual \\\hline
|
||||
$23$ & $42$ & $1{,}970$ & $0$ \\
|
||||
$24$ & $44$ & $6{,}833$ & $1$ \quad (Holton--McKay Fig.~2.10) \\
|
||||
$25$ & $46$ & $23{,}384$ & $1$ \\
|
||||
$26$ & $48$ & $82{,}625$ & $0$ \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
The $n = 23$ row recomputes Faulkner--Younger's minimality (no cyclically
|
||||
$5$-connected non-Hamiltonian cubic planar graph below $44$ vertices). At
|
||||
$n = 25$ we find a single new cyclically $5$-connected non-Hamiltonian
|
||||
cubic planar graph on $46$ vertices; its dual we call $T_{25}$, a
|
||||
$25$-vertex $5$-connected triangulation with degree sequence
|
||||
$5^{21}\,8^{3}\,9^{1}$. At $n = 26$ there are again none. (All counts
|
||||
depend on the correctness of \texttt{plantri} and of the Hamiltonicity
|
||||
test.)
|
||||
|
||||
An exhaustive scan of all $\binom{n}{5}$ candidate $5$-cuts confirms that
|
||||
both $T$ and $T_{25}$ are \emph{internally $6$-connected}: every
|
||||
$5$-element vertex cut is the neighbourhood of a single degree-$5$
|
||||
vertex, so neither admits a nontrivial separation of size $\le 5$. This
|
||||
is the strongest connectivity a planar triangulation can have -- planar
|
||||
graphs are never $6$-connected, because every planar graph has a vertex
|
||||
of degree $\le 5$ -- and it is the level at which Birkhoff-style
|
||||
reductions terminate (a minimal counterexample to the four colour theorem
|
||||
can be assumed internally $6$-connected). Both cores are therefore
|
||||
genuinely irreducible bases of any decomposition-based argument: nothing
|
||||
in the cut-decomposition can simplify them further.
|
||||
|
||||
The bridge-derivability test on $T_{25}$ follows the same recipe and
|
||||
yields the same conclusion. Enumerating valid parity partitions and
|
||||
ordering by total Betti, the minimum total Betti is $1$, and the very
|
||||
first Betti-$1$ partition encountered with a small backward bridge-orbit
|
||||
contains an Even Level Graph (source $s = 24$, maximum level $4$) at
|
||||
depth $2$ in an orbit of $3{,}114$ states. The two bridge switches
|
||||
carrying that Even Level Graph to $T_{25}$ are
|
||||
\[
|
||||
\text{remove } \{21,23\},\ \text{add } \{22,24\}
|
||||
\quad\text{and}\quad
|
||||
\text{remove } \{3,5\},\ \text{add } \{1,6\},
|
||||
\]
|
||||
each adding a same-parity edge that is a bridge of the even parity
|
||||
subgraph; both steps are valid bridge switches
|
||||
(Figure~\ref{fig:n25-dual}). The witness signature -- minimum total
|
||||
Betti, a tiny bridge-orbit, a depth-$2$ Even Level Graph -- is the same
|
||||
that worked on $T$, suggesting it is not an accident of the $n = 24$
|
||||
example but the generic shape of bridge-derivability witnesses on
|
||||
internally-$6$-connected non-Hamiltonian-dual cores. The conjecture thus
|
||||
survives every irreducible small case the connectivity bound forces us
|
||||
to face.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{figures/core_n25_dual.png}
|
||||
\caption{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.}
|
||||
\label{fig:n25-dual}
|
||||
\end{figure}
|
||||
|
||||
\subsection*{Toward a characterization of bridge-derived graphs}
|
||||
|
||||
A bridge switch is a diagonal flip of the quadrilateral around a level
|
||||
|
||||
Reference in New Issue
Block a user