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 <noreply@anthropic.com>
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@@ -190,13 +190,19 @@ def check_clause_4(H, edges, col_list, named, witness):
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"""Construct the modified H + recoloring, identify f_n, check clause 4."""
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e1, e2 = witness['e1'], witness['e2']
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e_F = witness['e_F']
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a = col_list[e1] # color of e_1 = e_2
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a = col_list[e1] # color of e_1 = e_2 (clauses 1)
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merged_idx = edge_idx(edges, named['merged'])
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cyc_a, cyc_b = witness['kc_color_pair'] # = (c_merged, c_other)
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# a is the colour of e_1, e_2 on the Kempe cycle. By chord-apex,
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# c_merged is the color of merged = the color of all "a"-edges on the
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# cycle. The cycle alternates between c_merged = a and c_other = b.
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b = cyc_b
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# On the {cyc_a, cyc_b}-Kempe cycle, e_1 carries colour a, which
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# equals either cyc_a or cyc_b. The conjecture's "b" is the OTHER
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# colour on the cycle; we set it accordingly.
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if a == cyc_a:
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b = cyc_b
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elif a == cyc_b:
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b = cyc_a
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else:
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raise RuntimeError(
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f"e_1 colour {a} not in Kempe pair {(cyc_a, cyc_b)}")
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c = ({0, 1, 2} - {a, b}).pop()
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# Subdivided cycle K' alternates blue/green starting from merged = blue
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+131
@@ -0,0 +1,131 @@
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"""Test the face-monochromatic-pair conjecture (both the weak form,
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clauses 1-3, and the strengthening, clauses 1-4) on the 6 duals of the
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non-Hamiltonian 38-vertex cubic plane graphs found by Holton & McKay.
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Each Holton-McKay graph is itself a cubic plane graph $G'$; its dual is a
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21-vertex triangulation $G$. We treat each Holton-McKay graph as the $G'$
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of a hypothetical minimal counterexample and apply the reduced-dual
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construction at each of its pentagonal faces.
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Reuses the testing infrastructure from check_conj_3_8_scaled.py.
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Run with: sage experiments/check_conj_on_holton_mckay.py
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"""
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import os
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import sys
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from sage.all import Graph
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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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find_all_36_witnesses,
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check_clause_4,
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)
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from check_conj_face_kempe_scaled import conjecture_holds_for
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HM_FILE = ('/Users/didericis/Code/math-research/papers/'
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'even_level_graph_generators/experiments/nonham38m4.pc')
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def parse_planar_code_to_sage(path):
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"""Parse McKay's planar_code file and return list of Sage Graphs (each
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a 3-connected cubic plane graph)."""
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with open(path, 'rb') as f:
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data = f.read()
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header = b'>>planar_code<<'
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assert data.startswith(header), data[:20]
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pos = len(header)
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sage_graphs = []
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while pos < len(data):
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n = data[pos]; pos += 1
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edges = set()
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for v in range(n):
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while True:
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w = data[pos]; pos += 1
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if w == 0:
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break
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edges.add(frozenset((v, w - 1)))
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G = Graph([tuple(e) for e in edges], multiedges=False, loops=False)
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G.is_planar(set_embedding=True)
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sage_graphs.append(G)
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return sage_graphs
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def test_one(D, name=""):
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"""Test conjecture on one cubic plane graph $D$ ($= G'$). Returns
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(n_cand, n_sat_123, n_sat_1234, faces_used, indices_used)."""
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D.is_planar(set_embedding=True)
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n_cand = n_sat_123 = n_sat_1234 = 0
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face_count = 0
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for face in D.faces():
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if len(face) != 5:
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continue
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face_count += 1
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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:
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continue
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H = res['H']
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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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n_cand += 1
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if conjecture_holds_for(H, edges, col, named):
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n_sat_123 += 1
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# clauses 1-4: existential at the witness level
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witnesses = find_all_36_witnesses(H, edges, col, named)
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if any(check_clause_4(H, edges, col, named, w)
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for w in witnesses):
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n_sat_1234 += 1
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return n_cand, n_sat_123, n_sat_1234, face_count
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def main():
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graphs_list = parse_planar_code_to_sage(HM_FILE)
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print(f"Loaded {len(graphs_list)} Holton-McKay graphs from "
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f"{HM_FILE}\n")
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grand_cand = grand_123 = grand_1234 = 0
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rows = []
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import time
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for i, G in enumerate(graphs_list):
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n = G.order()
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m = G.size()
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degs = sorted(set(G.degree()))
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t0 = time.time()
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n_cand, n_123, n_1234, n_pent = test_one(G, name=f"HM{i}")
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dt = time.time() - t0
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rows.append((i, n, m, degs, n_pent, n_cand, n_123, n_1234, dt))
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grand_cand += n_cand
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grand_123 += n_123
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grand_1234 += n_1234
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sys_stdout = sys.stdout
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print(f"HM{i}: n={n} m={m} deg={degs} pent_faces={n_pent} "
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f"cand={n_cand} sat(1-3)={n_123} sat(1-4)={n_1234} "
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f"[{dt:.1f}s]")
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sys.stdout.flush()
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print()
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print("=" * 78)
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print(f"{'i':>2} {'n':>3} {'m':>4} {'#pent':>6} {'#cand':>8} "
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f"{'#sat(1-3)':>10} {'#sat(1-4)':>10} {'time':>7}")
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print("-" * 78)
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for i, n, m, degs, n_pent, n_cand, n_123, n_1234, dt in rows:
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print(f"{i:>2} {n:>3} {m:>4} {n_pent:>6} {n_cand:>8} "
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f"{n_123:>10} {n_1234:>10} {dt:>6.1f}s")
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print("-" * 78)
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print(f"{'tot':>2} {'-':>3} {'-':>4} {'-':>6} {grand_cand:>8} "
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f"{grand_123:>10} {grand_1234:>10}")
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if __name__ == '__main__':
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main()
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@@ -1,4 +1,4 @@
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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
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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
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entering extended mode
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restricted \write18 enabled.
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@@ -310,7 +310,7 @@ cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/loc
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al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local
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/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/
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texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
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Output written on paper.pdf (12 pages, 1015233 bytes).
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Output written on paper.pdf (12 pages, 1017869 bytes).
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@@ -745,6 +745,31 @@ Conjecture-\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}-witnesses
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individually satisfy clause~(4) on each colouring, but in every case
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\emph{some} witness does. Clause~(4) is therefore an existential statement
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at the witness level, not a property of every witness.
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\smallskip\noindent\emph{Targeted check on the Holton--McKay duals.}
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The six $21$-vertex triangulations whose duals are the
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non-Hamiltonian $38$-vertex cubic plane graphs of Holton and McKay
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(the smallest examples falsifying Tait's conjecture) are a particularly
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interesting subfamily at $n = 21$. Running the strengthened test directly
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on these six (see
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\texttt{experiments/check\_conj\_on\_holton\_mckay.py}) gives:
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\begin{center}
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\small
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\renewcommand{\arraystretch}{1.15}
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\begin{tabular}{r|r|r|r|l}
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HM\# & \#pentagonal faces & \#col.\ tested & \#sat.\ (1)--(4) & status \\
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\hline
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$0$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
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$1$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
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$2$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
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$3$ & $10$ & $2{,}880$ & $2{,}880$ & all pass \\
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$4$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
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$5$ & $11$ & $2{,}880$ & $2{,}880$ & all pass \\
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\hline
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total & --- & $17{,}280$ & $17{,}280$ & \\
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\end{tabular}
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\end{center}
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\end{remark}
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\begin{remark}[The implication to the Four Colour Theorem]
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