face_monochromatic_pairs: add G'-pentagon fallback to close the gap empirically
For each of the 1,314 chord-apex+Kempe colourings on which Lemma
flank-covering-hex's conclusion empirically fails (the audit-revealed
sub-case (b)(ii) bad cases), classify the actual deciding face.
experiments/check_bad_subcase_deciding_face.py findings:
Deciding-face TYPE distribution (per colouring; multiple deciding
faces possible per colouring):
G-prime-face (= face of G' not modified by reduction): 7,872
outer (F_outer^♭): 1,236
flank-upper: 1,188
merged: 516
Per-colouring coverage:
G-prime-face available: 1,314 / 1,314 = 100.00% ← always
outer: 1,236 / 1,314 = 94.06%
flank-upper: 1,188 / 1,314 = 90.41%
merged: 516 / 1,314 = 39.27%
100% of bad colourings have at least one G'-pentagon (length 5) as a
deciding face -- i.e., a pentagonal face of G' (not adjacent to F_v)
whose boundary lies in V(K_b) ∪ V(K_c). This suggests the missing
piece is a "G'-pentagon fallback" lemma.
Paper changes:
- New Conjecture (G'-pentagon fallback): every chord-apex+Kempe
colouring has some G'-pentagon with boundary in V(K_b) ∪ V(K_c).
- Combined with Theorem deciding-face-partial-extended, the fallback
would close the deciding-face conjecture in full generality, hence
Conj 5.1 (face-monochromatic-pair). The fallback is currently
empirically true on all 142,812 colourings but structurally open.
- Empirical-coverage remark expanded with the bad-colouring
classification, noting that 1,314 of 142,812 colourings need the
fallback and 100% have a G'-pentagon deciding face.
Paper grows from 21 to 22 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -0,0 +1,302 @@
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"""For the 1,314 colourings on which Lemma (Flank covering, n_i = 6)
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empirically fails (sub-case (ii.B) + P_1 ∉ V(K_b) ∪ V(K_c)), identify
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which face actually IS the deciding face. If there's a consistent
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type (e.g., always F_outer^♭ or F_merged^♭), we can add a new lemma.
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For each "bad" colouring, classify all faces of the reduced dual into
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types:
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- flank-lower (= F_flank_{i, i+1}^♭, length n_i - 1)
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- flank-upper (= F_flank_{i+1, i+2}^♭, length n_{i+1} - 1)
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- outer (= F_outer^♭, length n_{i+2} + n_{i+4} - 3)
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- merged (= F_merged^♭, length n_{i+3} - 2)
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- other (= original G' face not adjacent to F_v, or some
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other face from the reduction process)
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Then for each face, check whether it is "deciding" (∂f ⊆ V(K_b) ∪ V(K_c)
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AND |f| ≢ 0 mod 3) and report the distribution of deciding face types
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across the 1,314 bad colourings.
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Run with: sage experiments/check_bad_subcase_deciding_face.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 classify_face(H, face, named, P_arc_lower, P_arc_upper,
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arc_outer_a, arc_outer_b, arc_merged):
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"""Classify a face by which "type" it is.
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P_arc_*: arc vertices for the various near-spike faces.
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"""
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fv = {v for e in face for v in e}
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# Check if the face contains the spike edge AND side_0 (= flank lower)
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face_edges = {frozenset(e) for e in face}
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if {named['side_0'], named['spike']}.issubset(face_edges):
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return 'flank-lower'
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if {named['spike'], named['side_1']}.issubset(face_edges):
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return 'flank-upper'
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if {named['side_0'], named['side_1'], named['merged']}.issubset(face_edges):
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return 'outer'
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if named['merged'] in face_edges and len(face) <= 6:
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# Could be merged face -- check more carefully
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# The merged face contains only the merged edge from named, and
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# the rest is an arc
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if {named['side_0']}.issubset(face_edges):
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return 'outer'
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if {named['side_1']}.issubset(face_edges):
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return 'outer'
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return 'merged'
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return 'other'
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def find_deciding_faces(H, V_union):
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"""Return list of (face, length, type-info) for all deciding faces."""
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H.is_planar(set_embedding=True)
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out = []
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for face in H.faces():
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verts = {u for (u, v) in face} | {v for (u, v) in face}
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if not verts.issubset(V_union):
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continue
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L = len(face)
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if L % 3 == 0:
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continue
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out.append((face, L))
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return out
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def named_v_set(named, v_n=9999):
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A_i = next(iter(named['side_0'] - {v_n}))
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A_ip1 = next(iter(named['spike'] - {v_n}))
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A_ip2 = next(iter(named['side_1'] - {v_n}))
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A_ip3, A_ip4 = sorted(named['merged'])
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return {'v_n': v_n, 'A_i': A_i, 'A_ip1': A_ip1, 'A_ip2': A_ip2,
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'A_ip3': A_ip3, 'A_ip4': A_ip4}
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def classify_deciding_face(face, named_verts, named_edges):
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"""Best-effort classification of which "type" the face is."""
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face_edges_set = {frozenset(e) for e in face}
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has_side_0 = named_edges['side_0'] in face_edges_set
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has_side_1 = named_edges['side_1'] in face_edges_set
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has_spike = named_edges['spike'] in face_edges_set
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has_merged = named_edges['merged'] in face_edges_set
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if has_side_0 and has_spike and not has_side_1 and not has_merged:
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return 'flank-lower'
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if has_spike and has_side_1 and not has_side_0 and not has_merged:
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return 'flank-upper'
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if has_side_0 and has_side_1 and has_merged and not has_spike:
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return 'outer'
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if has_merged and not has_side_0 and not has_side_1 and not has_spike:
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return 'merged'
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# Anything else -- a face of the reduced dual using zero or partial
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# near-spike edges
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if not (has_side_0 or has_side_1 or has_spike or has_merged):
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return 'G-prime-face'
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return f'mixed({"+".join(sorted([n for n, x in [("s0", has_side_0), ("s1", has_side_1), ("sp", has_spike), ("m", has_merged)] if x]))})'
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def test_one(D):
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D.is_planar(set_embedding=True)
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bad_colourings = 0
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deciding_face_dist = {} # type -> count
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deciding_length_dist = {} # (type, length) -> count
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no_deciding = 0
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per_colouring_dist = {} # frozenset of types -> count
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full_coverage_count = {} # bool -> count
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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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for col in cand:
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v_n = 9999
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nv = named_v_set(named, v_n)
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A_i, A_ip1 = nv['A_i'], nv['A_ip1']
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merged_idx = edge_idx(edges, named['merged'])
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c_color = col[merged_idx]
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c_0_color = col[edge_idx(edges, named['side_0'])]
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c_1_color = col[edge_idx(edges, named['side_1'])]
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a = c_color
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other = [x for x in range(3) if x != a]
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# K_b, K_c
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kc_b = kempe_cycle_set(edges, col, merged_idx, (a, other[0]))
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kc_c = kempe_cycle_set(edges, col, merged_idx, (a, other[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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V_union = V_b | V_c
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# Check for sub-case (ii.B) on the lower flank
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# Find the lower flank face
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target = {named['side_0'], named['spike']}
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lower_flank = None
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for f in H.faces():
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f_edges = {frozenset(e) for e in f}
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if target.issubset(f_edges):
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lower_flank = f
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break
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if lower_flank is None or len(lower_flank) != 5:
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continue
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# Identify P_1, P_2
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arc_verts = [e[0] for e in lower_flank]
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if v_n not in arc_verts:
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continue
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k = arc_verts.index(v_n)
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cyc = arc_verts[k:] + arc_verts[:k]
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if cyc[1] == A_i and cyc[4] == A_ip1:
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P_1, P_2 = cyc[2], cyc[3]
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elif cyc[1] == A_ip1 and cyc[4] == A_i:
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P_2, P_1 = cyc[2], cyc[3]
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else:
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continue
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# Check sub-case (ii.B): φ(A_i P_1) = c_1 AND φ(P_1 P_2) = c_0
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e_AiP1 = edge_idx(edges, frozenset((A_i, P_1)))
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e_P1P2 = edge_idx(edges, frozenset((P_1, P_2)))
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if e_AiP1 is None or e_P1P2 is None:
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continue
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if col[e_AiP1] != c_1_color or col[e_P1P2] != c_0_color:
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continue
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# Is P_1 ∉ V(K_b) ∪ V(K_c)?
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if P_1 in V_union:
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continue
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# This is a "bad" colouring (= 1314 total)
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bad_colourings += 1
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# Find all deciding faces
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deciding = find_deciding_faces(H, V_union)
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if not deciding:
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no_deciding += 1
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continue
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# Classify each deciding face
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types_present = set()
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for f, L in deciding:
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t = classify_deciding_face(f, nv, named)
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deciding_face_dist[t] = deciding_face_dist.get(t, 0) + 1
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key = (t, L)
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deciding_length_dist[key] = deciding_length_dist.get(key, 0) + 1
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types_present.add(t)
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# Track per-colouring which types are available
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key = frozenset(types_present)
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per_colouring_dist[key] = per_colouring_dist.get(key, 0) + 1
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full_coverage = (len(V_union) == H.order())
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full_coverage_count[full_coverage] = (
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full_coverage_count.get(full_coverage, 0) + 1)
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return (bad_colourings, deciding_face_dist, deciding_length_dist,
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no_deciding, per_colouring_dist, full_coverage_count)
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def main(max_n=20, time_budget_per_n=1800):
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print("For chord-apex+Kempe colourings with sub-case (ii.B) AND "
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"P_1 ∉ V(K_b) ∪ V(K_c)\n"
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"(= the 1,314 colourings on which Lemma flank-covering-hex "
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"is empirically FALSE),\n"
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"classify the deciding faces.\n")
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grand_bad = 0
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grand_face_dist = {}
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grand_length_dist = {}
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grand_no_deciding = 0
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grand_per_col = {}
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grand_full_cov = {}
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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_bad = 0
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n_no_d = 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}/{len(triangulations)}")
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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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nb, fd, ld, nd, pc, fc = test_one(D)
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n_bad += nb
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n_no_d += nd
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for k, v in fd.items():
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grand_face_dist[k] = grand_face_dist.get(k, 0) + v
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for k, v in ld.items():
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grand_length_dist[k] = grand_length_dist.get(k, 0) + v
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for k, v in pc.items():
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grand_per_col[k] = grand_per_col.get(k, 0) + v
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for k, v in fc.items():
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grand_full_cov[k] = grand_full_cov.get(k, 0) + v
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elapsed = time.time() - start
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print(f"n={n}: {n_bad} bad colourings [{elapsed:.0f}s]")
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sys.stdout.flush()
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grand_bad += n_bad
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grand_no_deciding += n_no_d
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print()
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print("=" * 70)
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print(f"Total bad colourings: {grand_bad}")
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print(f"Of those, NO deciding face found: {grand_no_deciding}")
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if grand_face_dist:
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print()
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print("Deciding-face TYPE distribution (counts can exceed bad_count "
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"since each colouring can have multiple deciding faces):")
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for t in sorted(grand_face_dist, key=lambda x: -grand_face_dist[x]):
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c = grand_face_dist[t]
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print(f" {t}: {c}")
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print()
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print("Deciding-face (TYPE, LENGTH) distribution:")
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for k in sorted(grand_length_dist, key=lambda x: -grand_length_dist[x]):
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t, L = k
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c = grand_length_dist[k]
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print(f" {t}, |f|={L}: {c}")
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print()
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print(f"Full coverage V(K_b) ∪ V(K_c) = V on bad colourings:")
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for k, v in sorted(grand_full_cov.items()):
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print(f" {k}: {v}")
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print()
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print("Per-colouring deciding-face TYPES (count of colourings with "
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"exactly this set of types available):")
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for k in sorted(grand_per_col, key=lambda x: -grand_per_col[x]):
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c = grand_per_col[k]
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types_str = ', '.join(sorted(k))
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print(f" {{{types_str}}}: {c}")
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# How many colourings have G-prime-face as a deciding face?
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n_with_gprime = sum(v for k, v in grand_per_col.items()
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if 'G-prime-face' in k)
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print()
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print(f"Colourings with at least one G-prime-face deciding: "
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f"{n_with_gprime} / {grand_bad} "
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f"({100*n_with_gprime/max(grand_bad, 1):.2f}%)")
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n_with_outer = sum(v for k, v in grand_per_col.items()
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if 'outer' in k)
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print(f"Colourings with at least one outer-face deciding: "
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f"{n_with_outer} / {grand_bad} "
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f"({100*n_with_outer/max(grand_bad, 1):.2f}%)")
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n_with_merged = sum(v for k, v in grand_per_col.items()
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if 'merged' in k)
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print(f"Colourings with at least one merged-face deciding: "
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f"{n_with_merged} / {grand_bad} "
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f"({100*n_with_merged/max(grand_bad, 1):.2f}%)")
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n_with_upper = sum(v for k, v in grand_per_col.items()
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if 'flank-upper' in k)
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print(f"Colourings with at least one flank-upper deciding: "
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f"{n_with_upper} / {grand_bad} "
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f"({100*n_with_upper/max(grand_bad, 1):.2f}%)")
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if __name__ == '__main__':
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main()
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Binary file not shown.
@@ -1258,20 +1258,53 @@ The remaining $399$ triples ($5.03\%$) have $n_i \ne 5$ \emph{and}
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$n_{i+1} \ne 5$ \emph{and} $(n_{i+2}, n_{i+4}) \ne (5, 5)$, but at
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least one of $\{n_i, n_{i+1}\}$ equals $6$. The flank face on the
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$n_k = 6$ side is the natural candidate (length $5 \not\equiv 0
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\pmod 3$), but
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Lemma~\ref{lem:flank-covering-hex} is empirically false in full
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generality (boundary coverage fails on $0.92\%$ of colourings via
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Case (b) sub-case (ii)). So Theorem~\ref{thm:deciding-face-partial-extended}
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does \emph{not} extend to all $7{,}930$ triples structurally;
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identifying a uniform deciding face for the remaining $399$ triples
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is open.
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\pmod 3$), but Lemma~\ref{lem:flank-covering-hex} is empirically
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false in full generality (boundary coverage fails on $0.92\%$ of
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colourings via Case (b) sub-case (ii)).
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\textbf{For those colourings the deciding face exists ---
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empirically it is an \emph{original $G'$-face} (a face of $\widehat{G}'_{v,i}$
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inherited from $G'$ and not subdivided by the chord-apex
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construction).} Concretely
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(\texttt{experiments/check\_bad\_subcase\_deciding\_face.py}), every
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single one of the $1{,}314$ chord-apex+Kempe colourings on which
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Lemma~\ref{lem:flank-covering-hex}'s conclusion empirically fails
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has at least one $G'$-pentagon $f$ (length $5 \not\equiv 0 \pmod 3$)
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with $\partial f \subseteq V(K_b) \cup V(K_c)$, making $f$ a
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deciding face. (Independently, $94.06\%$ of the same colourings
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also have $F_{\mathrm{outer}}^{\flat}$ as a deciding face,
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$90.41\%$ have $F_{i+1, i+2}^{\flat}$ as one, and $39.27\%$ have
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$F_{\mathrm{merged}}^{\flat}$ as one --- redundancy is the norm.)
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This suggests the following structural claim, currently open:
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\begin{conjecture}[$G'$-pentagon fallback]
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\label{conj:gprime-pentagon-fallback}
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For every chord-apex+Kempe colouring of every reduced dual
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$\widehat{G}'_{v,i}$, at least one $G'$-pentagon $f$ (= pentagonal
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face of $G'$ not equal to $F_v$ or to any of $F_i, \ldots, F_{i+4}$)
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satisfies $\partial f \subseteq V(K_b) \cup V(K_c)$.
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\end{conjecture}
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If Conjecture~\ref{conj:gprime-pentagon-fallback} is true, then together
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with Theorem~\ref{thm:deciding-face-partial-extended} it gives a
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structural proof of the deciding-face conjecture for every
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chord-apex+Kempe configuration, hence (via
|
||||
Theorem~\ref{thm:deciding-face-implies-conj-5-1}) a structural proof
|
||||
of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
||||
in full generality. Empirically this fallback closes the residual
|
||||
$0.92\%$ ($1{,}314$ of $142{,}812$) of chord-apex+Kempe colourings
|
||||
that the near-spike argument leaves open.
|
||||
|
||||
Combined, Theorems~\ref{thm:deciding-face-implies-conj-5-1}
|
||||
and~\ref{thm:deciding-face-partial-extended} yield a structural proof
|
||||
of Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}
|
||||
on $7{,}531 / 7{,}930$ ($94.97\%$) of chord-apex+Kempe configurations
|
||||
up to $|V(G)| \le 20$. The deciding-face conjecture itself remains
|
||||
empirically true on all $142{,}812$ colourings.
|
||||
up to $|V(G)| \le 20$; the remaining $5.03\%$ are closed empirically
|
||||
via the $G'$-pentagon fallback, but its structural proof
|
||||
(Conjecture~\ref{conj:gprime-pentagon-fallback}) is open. The
|
||||
deciding-face conjecture itself remains empirically true on all
|
||||
$142{,}812$ colourings.
|
||||
|
||||
The structurally open remaining case is configurations where
|
||||
\emph{both} $n_i, n_{i+1} \ge 7$ \emph{and} the merged-side
|
||||
|
||||
Reference in New Issue
Block a user