From 3d1b1eb4a398ec6d027e6cd7ddf0f3e1f13fa3ca Mon Sep 17 00:00:00 2001 From: didericis Date: Sun, 24 May 2026 14:45:56 -0400 Subject: [PATCH] dual_decomposition: verify strengthened conjecture on 6 Holton-McKay duals - New experiment experiments/check_conj_on_holton_mckay.py parses McKay's planar_code file of the 6 non-Hamiltonian 38-vertex cubic plane graphs (Holton-McKay) and tests both clauses (1)-(3) and (1)-(4) of the face-monochromatic-pair conjecture on each. Result: 17,280 candidate colourings, all 17,280 satisfy both conjectures. - Add a "Targeted check on the Holton-McKay duals" paragraph to Remark 4.4 with a per-graph table. - Fix a latent bug in check_conj_3_8_scaled.py: b was hardcoded to cyc_b, leaving b == a when phi(e_1) == cyc_b (and consequently c ambiguous). Now correctly computes b = whichever of cyc_a/cyc_b is not a, raising if neither matches. The bug never crashed n <= 20 because any() short-circuited on correctly-built witnesses; the Holton-McKay reductions hit it on the first witness, surfacing it. Co-Authored-By: Claude Opus 4.7 --- .../experiments/check_conj_3_8_scaled.py | 16 ++- .../experiments/check_conj_on_holton_mckay.py | 131 ++++++++++++++++++ .../paper.log | 4 +- .../paper.pdf | Bin 1015233 -> 1017869 bytes .../paper.tex | 25 ++++ 5 files changed, 169 insertions(+), 7 deletions(-) create mode 100644 papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_on_holton_mckay.py diff --git a/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_3_8_scaled.py b/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_3_8_scaled.py index 2a25b71..e0a9a15 100644 --- a/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_3_8_scaled.py +++ b/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_3_8_scaled.py @@ -190,13 +190,19 @@ def check_clause_4(H, edges, col_list, named, witness): """Construct the modified H + recoloring, identify f_n, check clause 4.""" e1, e2 = witness['e1'], witness['e2'] e_F = witness['e_F'] - a = col_list[e1] # color of e_1 = e_2 + a = col_list[e1] # color of e_1 = e_2 (clauses 1) merged_idx = edge_idx(edges, named['merged']) cyc_a, cyc_b = witness['kc_color_pair'] # = (c_merged, c_other) - # a is the colour of e_1, e_2 on the Kempe cycle. By chord-apex, - # c_merged is the color of merged = the color of all "a"-edges on the - # cycle. The cycle alternates between c_merged = a and c_other = b. - b = cyc_b + # On the {cyc_a, cyc_b}-Kempe cycle, e_1 carries colour a, which + # equals either cyc_a or cyc_b. The conjecture's "b" is the OTHER + # colour on the cycle; we set it accordingly. + if a == cyc_a: + b = cyc_b + elif a == cyc_b: + b = cyc_a + else: + raise RuntimeError( + f"e_1 colour {a} not in Kempe pair {(cyc_a, cyc_b)}") c = ({0, 1, 2} - {a, b}).pop() # Subdivided cycle K' alternates blue/green starting from merged = blue diff --git a/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_on_holton_mckay.py b/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_on_holton_mckay.py new file mode 100644 index 0000000..52d8500 --- /dev/null +++ b/papers/dual_decomposition_minimal_counterexamples/experiments/check_conj_on_holton_mckay.py @@ -0,0 +1,131 @@ +"""Test the face-monochromatic-pair conjecture (both the weak form, +clauses 1-3, and the strengthening, clauses 1-4) on the 6 duals of the +non-Hamiltonian 38-vertex cubic plane graphs found by Holton & McKay. + +Each Holton-McKay graph is itself a cubic plane graph $G'$; its dual is a +21-vertex triangulation $G$. We treat each Holton-McKay graph as the $G'$ +of a hypothetical minimal counterexample and apply the reduced-dual +construction at each of its pentagonal faces. + +Reuses the testing infrastructure from check_conj_3_8_scaled.py. + +Run with: sage experiments/check_conj_on_holton_mckay.py +""" +import os +import sys + +from sage.all import Graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) + +from check_conj_3_8_scaled import ( + apply_reduction, + proper_3_edge_colorings, + matches_chord_apex_kempe, + find_all_36_witnesses, + check_clause_4, +) +from check_conj_face_kempe_scaled import conjecture_holds_for + + +HM_FILE = ('/Users/didericis/Code/math-research/papers/' + 'even_level_graph_generators/experiments/nonham38m4.pc') + + +def parse_planar_code_to_sage(path): + """Parse McKay's planar_code file and return list of Sage Graphs (each + a 3-connected cubic plane graph).""" + with open(path, 'rb') as f: + data = f.read() + header = b'>>planar_code<<' + assert data.startswith(header), data[:20] + pos = len(header) + sage_graphs = [] + while pos < len(data): + n = data[pos]; pos += 1 + edges = set() + for v in range(n): + while True: + w = data[pos]; pos += 1 + if w == 0: + break + edges.add(frozenset((v, w - 1))) + G = Graph([tuple(e) for e in edges], multiedges=False, loops=False) + G.is_planar(set_embedding=True) + sage_graphs.append(G) + return sage_graphs + + +def test_one(D, name=""): + """Test conjecture on one cubic plane graph $D$ ($= G'$). Returns + (n_cand, n_sat_123, n_sat_1234, faces_used, indices_used).""" + D.is_planar(set_embedding=True) + n_cand = n_sat_123 = n_sat_1234 = 0 + face_count = 0 + for face in D.faces(): + if len(face) != 5: + continue + face_count += 1 + for i_red in range(5): + res = apply_reduction(D, face, i_red, 9999) + if res is None: + continue + H = res['H'] + named = res['named'] + H.is_planar(set_embedding=True) + edges, colorings = proper_3_edge_colorings(H) + cand = [c for c in colorings + if matches_chord_apex_kempe(edges, c, named)] + for col in cand: + n_cand += 1 + if conjecture_holds_for(H, edges, col, named): + n_sat_123 += 1 + # clauses 1-4: existential at the witness level + witnesses = find_all_36_witnesses(H, edges, col, named) + if any(check_clause_4(H, edges, col, named, w) + for w in witnesses): + n_sat_1234 += 1 + return n_cand, n_sat_123, n_sat_1234, face_count + + +def main(): + graphs_list = parse_planar_code_to_sage(HM_FILE) + print(f"Loaded {len(graphs_list)} Holton-McKay graphs from " + f"{HM_FILE}\n") + + grand_cand = grand_123 = grand_1234 = 0 + rows = [] + import time + for i, G in enumerate(graphs_list): + n = G.order() + m = G.size() + degs = sorted(set(G.degree())) + t0 = time.time() + n_cand, n_123, n_1234, n_pent = test_one(G, name=f"HM{i}") + dt = time.time() - t0 + rows.append((i, n, m, degs, n_pent, n_cand, n_123, n_1234, dt)) + grand_cand += n_cand + grand_123 += n_123 + grand_1234 += n_1234 + sys_stdout = sys.stdout + print(f"HM{i}: n={n} m={m} deg={degs} pent_faces={n_pent} " + f"cand={n_cand} sat(1-3)={n_123} sat(1-4)={n_1234} " + f"[{dt:.1f}s]") + sys.stdout.flush() + + print() + print("=" * 78) + print(f"{'i':>2} {'n':>3} {'m':>4} {'#pent':>6} {'#cand':>8} " + f"{'#sat(1-3)':>10} {'#sat(1-4)':>10} {'time':>7}") + print("-" * 78) + for i, n, m, degs, n_pent, n_cand, n_123, n_1234, dt in rows: + print(f"{i:>2} {n:>3} {m:>4} {n_pent:>6} {n_cand:>8} " + f"{n_123:>10} {n_1234:>10} {dt:>6.1f}s") + print("-" * 78) + print(f"{'tot':>2} {'-':>3} {'-':>4} {'-':>6} {grand_cand:>8} " + f"{grand_123:>10} {grand_1234:>10}") + + +if __name__ == '__main__': + main() diff --git a/papers/dual_decomposition_minimal_counterexamples/paper.log b/papers/dual_decomposition_minimal_counterexamples/paper.log index 9c74678..765488f 100644 --- a/papers/dual_decomposition_minimal_counterexamples/paper.log +++ b/papers/dual_decomposition_minimal_counterexamples/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:32 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 14:42 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -310,7 +310,7 @@ cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (12 pages, 1015233 bytes). +Output written on paper.pdf (12 pages, 1017869 bytes). 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