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master
| Author | SHA1 | Date | |
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 didericis
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f0fdae11d4 | ||
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didericis
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d9007c8697 |
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"""Survey: for each n, how many maximal planar graphs (plane-triangulation
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iso classes) are *bridge-derived* level graphs of some Even Level Graph.
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Bridge-derivedness is decided exhaustively via the backward bridge-switch
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orbit (see small_n_probe.is_bridge_derived): a triangulation G is
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bridge-derived iff some valid parity partition L of G admits an Even Level
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Graph (parity L) in G's backward bridge-orbit. Feasible only at small n.
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Also cross-tabulates against the intertwining-tree property so the two
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covering families in the disjunction conjecture can be compared.
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Usage: python3 bridge_derived_survey.py [n_max] (default 11)
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"""
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import sys
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import os
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import time
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sys.path.insert(0, '/Users/didericis/Code/math-research/papers/'
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'level_resolutions_of_maximal_planar_graphs/experiments')
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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from triangulation_gen import enumerate_all_triangulations
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from small_n_probe import is_bridge_derived
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from test_disjunction import is_intertwining_tree
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def survey_n(n):
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t0 = time.time()
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tris = enumerate_all_triangulations(n)
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n_bridge = 0
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n_inter = 0
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n_bridge_and_inter = 0
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n_bridge_only = 0
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n_inter_only = 0
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n_neither = 0
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for G in tris:
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bd = is_bridge_derived(G)
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it = is_intertwining_tree(G)[0]
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if bd:
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n_bridge += 1
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if it:
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n_inter += 1
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if bd and it:
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n_bridge_and_inter += 1
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elif bd:
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n_bridge_only += 1
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elif it:
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n_inter_only += 1
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else:
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n_neither += 1
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return {
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'n': n,
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'total': len(tris),
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'bridge': n_bridge,
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'inter': n_inter,
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'bridge_and_inter': n_bridge_and_inter,
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'bridge_only': n_bridge_only,
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'inter_only': n_inter_only,
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'neither': n_neither,
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'elapsed': time.time() - t0,
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}
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def main():
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n_max = int(sys.argv[1]) if len(sys.argv) > 1 else 11
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rows = []
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print(f'{"n":>3} {"total":>7} {"bridge-deriv":>13} {"%":>6} '
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f'{"inter":>7} {"b&i":>6} {"b-only":>7} {"i-only":>7} '
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f'{"neither":>8} {"sec":>7}', flush=True)
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for n in range(6, n_max + 1):
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r = survey_n(n)
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rows.append(r)
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pct = 100.0 * r['bridge'] / r['total'] if r['total'] else 0.0
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print(f'{r["n"]:>3} {r["total"]:>7} {r["bridge"]:>13} {pct:>5.1f}% '
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f'{r["inter"]:>7} {r["bridge_and_inter"]:>6} '
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f'{r["bridge_only"]:>7} {r["inter_only"]:>7} '
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f'{r["neither"]:>8} {r["elapsed"]:>6.1f}', flush=True)
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if r['neither']:
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print(f' *** {r["neither"]} triangulation(s) at n={n} are '
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f'NEITHER bridge-derived nor intertwining trees ***',
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flush=True)
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return rows
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if __name__ == '__main__':
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main()
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@@ -0,0 +1,167 @@
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"""Draw the two former "failures" (n=9 bipyramid, n=10 bipyramid+stacked) the
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RIGHT way: a proper 4-colouring whose 4 colours split into two complementary
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pairs, each inducing an outerplanar (bipartite => even-cycle) subgraph.
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Top row: the planar drawing with the 4-colouring; edges of the two split
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classes drawn in two styles. Bottom row: the two complementary subgraphs shown
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separately, each annotated outerplanar (tree / forest / even cycle).
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Renders into this experiments folder.
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"""
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import os
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from itertools import combinations
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import networkx as nx
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import matplotlib.pyplot as plt
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T24 = [(0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8),
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(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8),
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(2, 3), (2, 4), (3, 5), (4, 8), (5, 6), (6, 7), (7, 8)]
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T94 = [(0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9),
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(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8),
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(2, 3), (2, 4), (3, 5), (4, 8), (5, 6), (6, 7), (7, 8),
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(7, 9), (8, 9)]
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CMAP = {0: '#444444', 1: '#d62728', 2: '#1f77b4', 3: '#2ca02c'}
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CNAME = {0: 'grey', 1: 'red', 2: 'blue', 3: 'green'}
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SPLITS = [({0, 1}, {2, 3}), ({0, 2}, {1, 3}), ({0, 3}, {1, 2})]
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def is_outerplanar(G):
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if G.number_of_nodes() <= 3:
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return True
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H = G.copy()
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apex = max(H.nodes()) + 1
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for v in G.nodes():
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H.add_edge(apex, v)
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return nx.check_planarity(H)[0]
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def enumerate_4colorings(G):
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nodes = list(G.nodes())
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adj = {v: set(G.neighbors(v)) for v in nodes}
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coloring = {}
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def bt(i, mx):
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if i == len(nodes):
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yield dict(coloring)
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return
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v = nodes[i]
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used = {coloring[w] for w in adj[v] if w in coloring}
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for c in range(min(3, mx + 1) + 1):
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if c in used:
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continue
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coloring[v] = c
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yield from bt(i + 1, max(mx, c))
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del coloring[v]
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yield from bt(0, -1)
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def find_good_split(G):
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for col in enumerate_4colorings(G):
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for A, B in SPLITS:
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va = [v for v in G if col[v] in A]
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vb = [v for v in G if col[v] in B]
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GA, GB = G.subgraph(va), G.subgraph(vb)
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if (is_outerplanar(GA) and nx.is_bipartite(GA)
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and is_outerplanar(GB) and nx.is_bipartite(GB)):
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return col, (A, B)
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return None, None
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def bipyramid_pos(rim_cycle, apexA, apexB):
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k = len(rim_cycle)
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order = rim_cycle[1:] + [rim_cycle[0]]
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pos = {}
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for i, v in enumerate(order):
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x = 1.7 * (1 - 2 * i / (k - 1))
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y = 1.0 * (x / 1.7) ** 2
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pos[v] = (x, y)
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pos[apexA] = (0.0, 0.62)
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pos[apexB] = (0.0, -1.7)
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return pos
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def describe(G):
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if G.number_of_edges() == 0:
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return "edgeless (isolated vertices)"
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if nx.is_forest(G):
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return "forest (tree, no cycles)"
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girth_even = all(len(c) % 2 == 0 for c in nx.cycle_basis(G))
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comps = nx.number_connected_components(G)
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tag = "even cycles" if girth_even else "ODD CYCLE!"
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return f"outerplanar, {tag}, {comps} component(s)"
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def draw_case(fig, gs_row, edges, pos, title):
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G = nx.Graph(edges)
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col, split = find_good_split(G)
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A, B = split
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node_colors = [CMAP[col[v]] for v in G.nodes()]
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ea = [e for e in G.edges() if col[e[0]] in A and col[e[1]] in A]
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eb = [e for e in G.edges() if col[e[0]] in B and col[e[1]] in B]
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ecross = [e for e in G.edges() if e not in ea and e not in eb]
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# left: whole graph, both classes highlighted
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ax = fig.add_subplot(gs_row[0])
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nx.draw_networkx_edges(G, pos, ax=ax, edgelist=ecross,
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edge_color='#dddddd', width=1.0)
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nx.draw_networkx_edges(G, pos, ax=ax, edgelist=ea,
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edge_color='#e8860a', width=3.0)
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nx.draw_networkx_edges(G, pos, ax=ax, edgelist=eb,
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edge_color='#7b2fbf', width=3.0, style='dashed')
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nx.draw_networkx_nodes(G, pos, node_color=node_colors, node_size=720,
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edgecolors='black', linewidths=1.4, ax=ax)
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nx.draw_networkx_labels(G, pos, ax=ax, font_color='white',
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font_weight='bold', font_size=11)
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an = "/".join(CNAME[c] for c in sorted(A))
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bn = "/".join(CNAME[c] for c in sorted(B))
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ax.set_title(f"{title}\nsplit [{an}] (orange solid) | "
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f"[{bn}] (purple dashed)", fontsize=10)
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ax.axis('off'); ax.set_aspect('equal')
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# middle: subgraph A alone
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GA = G.subgraph([v for v in G if col[v] in A])
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axA = fig.add_subplot(gs_row[1])
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nx.draw_networkx_edges(GA, pos, ax=axA, edge_color='#e8860a', width=3.0)
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nx.draw_networkx_nodes(GA, pos, node_color=[CMAP[col[v]] for v in GA],
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node_size=720, edgecolors='black',
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linewidths=1.4, ax=axA)
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nx.draw_networkx_labels(GA, pos, ax=axA, font_color='white',
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font_weight='bold', font_size=11)
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axA.set_title(f"[{an}] subgraph\n{describe(GA)}", fontsize=9)
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axA.axis('off'); axA.set_aspect('equal')
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# right: subgraph B alone
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GB = G.subgraph([v for v in G if col[v] in B])
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axB = fig.add_subplot(gs_row[2])
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nx.draw_networkx_edges(GB, pos, ax=axB, edge_color='#7b2fbf', width=3.0)
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nx.draw_networkx_nodes(GB, pos, node_color=[CMAP[col[v]] for v in GB],
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node_size=720, edgecolors='black',
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linewidths=1.4, ax=axB)
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nx.draw_networkx_labels(GB, pos, ax=axB, font_color='white',
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font_weight='bold', font_size=11)
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axB.set_title(f"[{bn}] subgraph\n{describe(GB)}", fontsize=9)
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axB.axis('off'); axB.set_aspect('equal')
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rim = [2, 3, 5, 6, 7, 8, 4]
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pos24 = bipyramid_pos(rim, 0, 1)
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pos94 = dict(pos24)
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pos94[9] = ((pos24[0][0] + pos24[7][0] + pos24[8][0]) / 3,
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(pos24[0][1] + pos24[7][1] + pos24[8][1]) / 3)
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fig = plt.figure(figsize=(15, 10))
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gs = fig.add_gridspec(2, 3)
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draw_case(fig, [gs[0, 0], gs[0, 1], gs[0, 2]], T24, pos24,
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"n=9 T24: 7-gonal bipyramid")
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draw_case(fig, [gs[1, 0], gs[1, 1], gs[1, 2]], T94, pos94,
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"n=10 T94: bipyramid + stacked vertex 9")
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fig.suptitle("Both decompose: 4-colouring -> 2+2 colour split -> two "
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"complementary outerplanar even-cycle subgraphs", fontsize=12)
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fig.tight_layout(rect=[0, 0, 1, 0.97])
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out = os.path.join(os.path.dirname(os.path.abspath(__file__)),
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'split_decomposition.png')
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fig.savefig(out, dpi=140)
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print(f"wrote {out}")
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Binary file not shown.
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After Width: | Height: | Size: 255 KiB |
@@ -0,0 +1,125 @@
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"""Corrected sanity check for the nested-outerplanar-shells construction.
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The construction splits the FOUR colours into TWO complementary pairs (two
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parity classes). Each pair induces a bipartite subgraph; the two subgraphs
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partition the vertex set. For a nested-shell decomposition we want BOTH
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complementary subgraphs to be outerplanar (bipartite => only even cycles, so
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"even cycles" is automatic; outerplanar is the binding condition).
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There are exactly 3 ways to split {0,1,2,3} into two pairs:
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{0,1}|{2,3}, {0,2}|{1,3}, {0,3}|{1,2}.
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Criterion (per triangulation): does SOME proper 4-colouring admit SOME split
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whose two complementary subgraphs are both outerplanar?
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This is the right test (an earlier version wrongly demanded that all SIX
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colour pairs be outerplanar, which odd bipyramids fail on the apex/heavy-rim
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pair -- but that pair is never one we'd use).
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Usage: python3 two_color_split_survey.py [n_max] (default 10)
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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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import networkx as nx
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sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
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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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from triangulation_gen import enumerate_all_triangulations
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# the 3 ways to split 4 colours into two complementary pairs
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SPLITS = [(({0, 1}), ({2, 3})),
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(({0, 2}), ({1, 3})),
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(({0, 3}), ({1, 2}))]
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def is_outerplanar(G):
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if G.number_of_nodes() <= 3:
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return True
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H = G.copy()
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apex = max(H.nodes()) + 1
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for v in G.nodes():
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H.add_edge(apex, v)
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return nx.check_planarity(H)[0]
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def enumerate_4colorings(G):
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nodes = list(G.nodes())
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adj = {v: set(G.neighbors(v)) for v in nodes}
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coloring = {}
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def bt(i, mx):
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if i == len(nodes):
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yield dict(coloring)
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return
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v = nodes[i]
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used = {coloring[w] for w in adj[v] if w in coloring}
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for c in range(min(3, mx + 1) + 1):
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if c in used:
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continue
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coloring[v] = c
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yield from bt(i + 1, max(mx, c))
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del coloring[v]
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yield from bt(0, -1)
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def outerplanar_even(G):
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"""Outerplanar AND every cycle even (i.e. bipartite). Disconnected ok.
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For a two-colour subgraph of a proper colouring bipartiteness is automatic,
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but we verify it explicitly so the criterion is honestly enforced."""
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return is_outerplanar(G) and nx.is_bipartite(G)
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def good_split(G, col):
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"""Return the first (sideA, sideB) split whose two complementary
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subgraphs are both outerplanar-with-even-cycles, or None.
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Disconnected subgraphs are allowed."""
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for A, B in SPLITS:
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va = [v for v in G if col[v] in A]
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vb = [v for v in G if col[v] in B]
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if outerplanar_even(G.subgraph(va)) and outerplanar_even(G.subgraph(vb)):
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return (A, B)
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return None
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def has_good_coloring(G):
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"""True iff SOME proper 4-colouring admits a valid 2+2 split.
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Early-exits on the first good colouring."""
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for col in enumerate_4colorings(G):
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if good_split(G, col) is not None:
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return True
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return False
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def survey_n(n):
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t0 = time.time()
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tris = enumerate_all_triangulations(n)
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n_good = 0
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bad = []
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for gi, G in enumerate(tris):
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ok = has_good_coloring(G)
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if ok:
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n_good += 1
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else:
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bad.append((gi, G))
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return n, len(tris), n_good, bad, time.time() - t0
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def main():
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n_max = int(sys.argv[1]) if len(sys.argv) > 1 else 10
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print(f"{'n':>3} {'tris':>6} {'has 2+2 split':>14} {'time(s)':>8}")
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print("-" * 38)
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for n in range(6, n_max + 1):
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n, ntri, ngood, bad, dt = survey_n(n)
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flag = "" if ngood == ntri else " <-- GAP"
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print(f"{n:>3} {ntri:>6} {ngood:>9}/{ntri:<4} {dt:>8.1f}{flag}")
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for gi, G in bad[:5]:
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edges = sorted(tuple(sorted(e)) for e in G.edges())
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print(f" no split: T{gi} edges={edges}")
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if __name__ == "__main__":
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main()
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@@ -44,19 +44,21 @@
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\newlabel{def:intertwining-tree}{{4.6}{7}{Intertwining tree}{theorem.4.6}{}}
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\newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{7}{}{theorem.4.7}{}}
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\newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}}
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\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent }
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\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here.}}{8}{table.2}\protected@file@percent }
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\newlabel{tab:bridge-census}{{2}{8}{Bridge-derived census for $6 \leq n \leq 10$. \emph {bridge-derived} counts plane-triangulation iso classes that are bridge-derived level graphs of some Even Level Graph, decided by exhaustive backward bridge-switch search over all valid parity partitions; \% is its fraction of all triangulations. \emph {intertwining only} counts those that are intertwining trees but not bridge-derived; \emph {neither} counts those covered by no disjunct. Every triangulation in this range is an intertwining tree, and every bridge-derived one is too, so bridge-derived $\subseteq $ intertwining tree here}{table.2}{}}
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\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{tab:n21}{{3}{9}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.3}{}}
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Output written on paper.pdf (13 pages, 1386477 bytes).
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@@ -492,6 +492,45 @@ $n = 21$ and there are exactly $6$ of them. Below $n = 21$ every
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maximal planar graph is an intertwining tree, which is why the
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disjunction holds trivially in that range.
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The intertwining-tree disjunct therefore carries the conjecture by itself
|
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for all small $n$, but this leaves open how much of the load the
|
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bridge-derived disjunct is independently able to bear. To measure that we
|
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classified every triangulation at $6 \leq n \leq 10$ as bridge-derived or
|
||||
not, deciding bridge-derivedness exhaustively: a triangulation is
|
||||
bridge-derived iff some valid parity partition admits an Even Level Graph
|
||||
in its backward bridge-switch orbit (a search feasible only at these
|
||||
sizes). Table~\ref{tab:bridge-census} records the result. Three features
|
||||
stand out. First, the bridge-derived disjunct is substantive but far from
|
||||
universal on its own: its share of all triangulations falls steadily, from
|
||||
all of them at $n = 6$ to under two-thirds by $n = 10$. Second, the
|
||||
disjunction never relies on it in this range -- the \emph{intertwining
|
||||
only} column counts triangulations covered by the tree disjunct alone, and
|
||||
it grows, while no triangulation here is bridge-derived without also being
|
||||
an intertwining tree. Third, and consistent with the conjecture, the
|
||||
\emph{neither} column is identically zero throughout.
|
||||
|
||||
\begin{table}[ht]
|
||||
\centering
|
||||
\begin{tabular}{cccccc}
|
||||
$n$ & triangulations & bridge-derived & \% & intertwining only & neither \\\hline
|
||||
$6$ & $2$ & $2$ & $100.0$ & $0$ & $0$ \\
|
||||
$7$ & $5$ & $4$ & $80.0$ & $1$ & $0$ \\
|
||||
$8$ & $14$ & $12$ & $85.7$ & $2$ & $0$ \\
|
||||
$9$ & $50$ & $36$ & $72.0$ & $14$ & $0$ \\
|
||||
$10$ & $233$ & $146$ & $62.7$ & $87$ & $0$ \\
|
||||
\end{tabular}
|
||||
\caption{Bridge-derived census for $6 \leq n \leq 10$. \emph{bridge-derived}
|
||||
counts plane-triangulation iso classes that are bridge-derived level graphs
|
||||
of some Even Level Graph, decided by exhaustive backward bridge-switch
|
||||
search over all valid parity partitions; \% is its fraction of all
|
||||
triangulations. \emph{intertwining only} counts those that are intertwining
|
||||
trees but not bridge-derived; \emph{neither} counts those covered by no
|
||||
disjunct. Every triangulation in this range is an intertwining tree, and
|
||||
every bridge-derived one is too, so bridge-derived $\subseteq$ intertwining
|
||||
tree here.}
|
||||
\label{tab:bridge-census}
|
||||
\end{table}
|
||||
|
||||
\subsection*{The boundary case $n = 21$}
|
||||
|
||||
The first triangulations that are \emph{not} intertwining trees are the
|
||||
|
||||
Reference in New Issue
Block a user