face_monochromatic_pairs: investigate (|S|, # pent F_k) joint distribution
experiments/check_S_vs_pent_Fk.py: joint distribution of |S| and
# pentagonal F_k (= count of n_i, n_{i+1}, n_{i+3} = 5, i.e.
"visible" pent F_k via flank/merged faces).
Across all 142,812 chord-apex+Kempe colourings:
- |S| = 0 dominates: 73.9% have full coverage.
- For |S| = 2, 4, 6, 8: distribution of visible pent F_k spans 0-3
with no clean monotone trend.
- |S| = 12, 14 cases NEVER have visible = 0 (= 0 occurrences in
the 0-column for these |S| values).
- The 30 special "|S|=8 hit=8" cases all have full p_G = 11 (= all
5 of v's neighbours degree ≥ 6), not just visible = 0.
So the obvious |S| ↔ # pent F_k coupling doesn't hold uniformly.
The "|S|=8 hit=8 ⇒ p_G = 11" empirical fact is specific to the
conjunction (high |S| + high hit), not to |S| alone.
For a structural proof of "|S|=8 + hit=8 ⇒ p_G = 11", we'd need a
deep Kempe-cycle-structural argument that hitting 8 G'-pentagons
with an 8-vertex S-cycle requires specific local geometry around v.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""Empirical investigation: for each chord-apex+Kempe colouring (NOT
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restricted to bad ones), correlate |S| = |V \\ (V(K_b) ∪ V(K_c))|
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with the number of pentagonal F_k adjacent to F_v in the parent G'.
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Hypothesis (from the |S|=8 hit=8 finding): there's a coupling where
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larger |S| forces fewer pentagonal F_k.
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Concretely: for each colouring, record:
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- |S|
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- # pentagonal F_k (= # of v's 5 neighbours in G with degree 5)
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- # G'-pentagons hit by S (when applicable)
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Aggregate the joint distribution.
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Run with: sage experiments/check_S_vs_pent_Fk.py
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"""
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import os
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import sys
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import time
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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kempe_cycle_set,
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edge_idx,
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)
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from check_heawood_on_kempe import dual_of, vertices_of_kempe
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def cyclic_neighbour_degrees(G, v):
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G.is_planar(set_embedding=True)
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emb = G.get_embedding()
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return [G.degree(u) for u in emb[v]]
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def test_one(D, G_parent):
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D.is_planar(set_embedding=True)
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G_parent.is_planar(set_embedding=True)
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joint_dist = {} # (|S|, # pent F_k) -> count
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for v_parent in G_parent.vertex_iterator():
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if G_parent.degree(v_parent) != 5: continue
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cyc_degs = cyclic_neighbour_degrees(G_parent, v_parent)
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n_pent_Fk_all = sum(1 for d in cyc_degs if d == 5)
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# For each reduction index i (0 to 4), determine the specific
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# pentagonal F_k among (F_0, ..., F_4) (= all 5).
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# We need to map v_parent + i_red to a specific reduction.
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# Use a different approach: enumerate reductions and check.
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# For each pentagonal face F_v of D (= matches v_parent),
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# iterate i_red.
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# Find D-faces matching v_parent's incident G'-faces:
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# (this is harder without explicit dual labelling)
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pass
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# Alternative approach: iterate over D's faces and reductions
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for face in D.faces():
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if len(face) != 5: continue
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for i_red in range(5):
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res = apply_reduction(D, face, i_red, 9999)
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if res is None: continue
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H = res['H']; named = res['named']
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H.is_planar(set_embedding=True)
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edges, colorings = proper_3_edge_colorings(H)
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cand = [c for c in colorings
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if matches_chord_apex_kempe(edges, c, named)]
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v_n = 9999
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# Compute n_k from face lengths in reduced dual
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n_pent_Fk = 0
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for f in H.faces():
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fset = {frozenset(e) for e in f}
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# F_flank_lower = len n_i - 1; if = 4, n_i = 5 (pentagonal).
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if named['side_0'] in fset and named['spike'] in fset:
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if len(f) == 4:
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n_pent_Fk += 1
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if named['spike'] in fset and named['side_1'] in fset:
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if len(f) == 4:
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n_pent_Fk += 1
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if (named['side_0'] in fset and named['side_1'] in fset
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and named['merged'] in fset):
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# F_outer = n_{i+2} + n_{i+4} - 3
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# For F_outer = 7 = 5+5-3, both n_{i+2}, n_{i+4} = 5.
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# For F_outer = 8 = 5+6-3 or 6+5-3, one = 5.
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# Hmm we can't tell which side from outer length alone.
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pass
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if (named['merged'] in fset and named['side_0'] not in fset
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and named['side_1'] not in fset
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and named['spike'] not in fset):
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# F_merged = n_{i+3} - 2
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if len(f) == 3:
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n_pent_Fk += 1
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# (Note: this counts pent F_k in n_i, n_{i+1}, n_{i+3} but
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# not in n_{i+2} or n_{i+4} which are mixed by F_outer.)
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# For better counting, use the parent graph.
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for col in cand:
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merged_idx = edge_idx(edges, named['merged'])
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a = col[merged_idx]
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bs = [c for c in range(3) if c != a]
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kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0]))
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kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1]))
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V_b = vertices_of_kempe(edges, kc_b)
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V_c = vertices_of_kempe(edges, kc_c)
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S_size = H.order() - len(V_b | V_c)
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key = (S_size, n_pent_Fk)
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joint_dist[key] = joint_dist.get(key, 0) + 1
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return joint_dist
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def main(max_n=20, time_budget_per_n=1800):
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print("Joint distribution of (|S|, # pentagonal F_k via flank/merged)\n"
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"across all chord-apex+Kempe colourings (NOT restricted to bad).\n"
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"\nNote: this counts pent F_k via flank-lower (= n_i = 5),\n"
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"flank-upper (= n_{i+1} = 5), and merged (= n_{i+3} = 5);\n"
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"it does NOT count pent F_k among n_{i+2}, n_{i+4} (= outer-side)\n"
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"which can't be distinguished from F_outer length alone.\n")
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grand_dist = {}
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for n in range(12, max_n + 1):
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start = time.time()
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try:
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triangulations = list(graphs.triangulations(n, minimum_degree=5))
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except Exception as ex:
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print(f"n={n}: cannot enumerate ({ex})")
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continue
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n_col = 0
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for tri_idx, G in enumerate(triangulations):
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if time.time() - start > time_budget_per_n:
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print(f" n={n}: timeout at tri {tri_idx}")
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break
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G.is_planar(set_embedding=True)
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D = dual_of(G)
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jd = test_one(D, G)
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for k, v in jd.items():
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grand_dist[k] = grand_dist.get(k, 0) + v
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n_col += v
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elapsed = time.time() - start
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print(f"n={n}: {n_col} colourings counted [{elapsed:.0f}s]")
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sys.stdout.flush()
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print()
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print("=" * 70)
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print("Joint distribution (|S|, # pent F_k visible):\n")
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# Print as a table
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all_S = sorted(set(k[0] for k in grand_dist))
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all_F = sorted(set(k[1] for k in grand_dist))
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print(f" |S|\\#pent_Fk_visible", end=' ')
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for f in all_F:
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print(f"{f:>7}", end='')
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print(f"{'total':>9}")
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for s in all_S:
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print(f" |S| = {s:>3} ", end=' ')
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total = 0
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for f in all_F:
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c = grand_dist.get((s, f), 0)
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total += c
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print(f"{c:>7}", end='')
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print(f"{total:>9}")
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print(f" total ", end=' ')
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for f in all_F:
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col_total = sum(grand_dist.get((s, f), 0) for s in all_S)
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print(f"{col_total:>7}", end='')
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grand_total = sum(grand_dist.values())
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print(f"{grand_total:>9}")
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if __name__ == '__main__':
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main()
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