math-research/papers/face_monochromatic_pairs/experiments/check_v_neighbour_degrees.py

"""For every reduction (G, v, i) underlying a chord-apex+Kempe
colouring (up to |V(G)| ≤ 20), check the degrees of v's five
neighbours in G.

This determines whether the partial structural proof of the
deciding-face conjecture (Theorem~\\ref{thm:deciding-face-partial} in
the paper) covers ALL configurations empirically:

  - If every (G, v) we encounter has at least one neighbour of v with
    deg_G(·) ≤ 6, the partial proof (which requires at least one
    n_k ∈ {5, 6}, equivalently at least one u_{k+1} with deg_G ≤ 6)
    handles every chord-apex+Kempe colouring up to |V(G)| ≤ 20.

  - If some (G, v) has all five neighbours of degree ≥ 7, we'd need
    the unproved merged-side argument for those.

Reports per n_G: number of (G, v) pairs with all-≥-7 neighbours.

Run with: sage experiments/check_v_neighbour_degrees.py
"""
import os
import sys
import time

from sage.all import Graph
from sage.graphs.graph_generators import graphs

HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)


def cyclic_neighbour_degrees(G, v):
    """Return the cyclic sequence of degrees of v's neighbours, in
    their CW order around v in G's planar embedding."""
    G.is_planar(set_embedding=True)
    emb = G.get_embedding()
    return [G.degree(u) for u in emb[v]]


def main(max_n=20, time_budget_per_n=1200):
    print("For every (G, v, i) underlying a chord-apex+Kempe colouring, "
          "check the\ndegrees of v's neighbours in G.\n")
    print("Each reduction index i ∈ {0,…,4} picks two consecutive "
          "neighbours\n(u_{i+1}, u_{i+2}) of v whose degrees become the\n"
          "two flank-face adjacent G'-face lengths n_i, n_{i+1}.\n")
    print(f"n_G in [12, {max_n}]\n")

    grand_triples = 0           # total (G, v, i) triples
    grand_bad_triples = 0       # both n_i, n_{i+1} ≥ 7
    grand_pairs = 0             # (G, v) pairs
    grand_all_ge_7 = 0          # (G, v) pairs with all 5 neighbours deg ≥ 7
    grand_dist_min = {}         # min neighbour deg over (G, v)
    grand_dist_consec = {}      # min over consecutive pair (n_i, n_{i+1}) for some i
    grand_examples = []

    for n in range(12, max_n + 1):
        start = time.time()
        try:
            triangulations = list(graphs.triangulations(n, minimum_degree=5))
        except Exception as ex:
            print(f"n={n}: cannot enumerate ({ex})")
            continue
        triples_n = 0
        bad_triples_n = 0
        pairs_n = 0
        all_ge_7_n = 0
        for tri_idx, G in enumerate(triangulations):
            if time.time() - start > time_budget_per_n:
                print(f"  n={n}: timeout at tri {tri_idx}/{len(triangulations)}")
                break
            for v in G.vertex_iterator():
                if G.degree(v) != 5:
                    continue
                cyc = cyclic_neighbour_degrees(G, v)
                pairs_n += 1
                m_all = min(cyc)
                grand_dist_min[m_all] = grand_dist_min.get(m_all, 0) + 1
                if m_all >= 7:
                    all_ge_7_n += 1
                worst_consec_min = max(min(cyc[i], cyc[(i+1) % 5])
                                       for i in range(5))
                grand_dist_consec[worst_consec_min] = (
                    grand_dist_consec.get(worst_consec_min, 0) + 1)
                for i in range(5):
                    triples_n += 1
                    n_i = cyc[(i + 1) % 5]
                    n_ip1 = cyc[(i + 2) % 5]
                    n_ip2 = cyc[(i + 3) % 5]
                    n_ip3 = cyc[(i + 4) % 5]
                    n_ip4 = cyc[(i + 5) % 5]
                    # AUDIT: does the structurally-tight subset (drop n_i=6 case) cover?
                    tight_covered = (
                        n_i == 5 or n_ip1 == 5 or
                        (n_ip2 == 5 and n_ip4 == 5)
                    )
                    if not tight_covered:
                        # Record what would be the "audit-uncovered" triples
                        # (= we have proved coverage only on n_i = 5 or n_{i+1} = 5
                        #   or (n_{i+2}, n_{i+4}) = (5, 5)).
                        pass  # counted below as "audit_uncovered_n"
                    if n_i >= 7 and n_ip1 >= 7:
                        bad_triples_n += 1
                        # Check whether F_outer's length condition is OK
                        outer_len = n_ip2 + n_ip4 - 3
                        outer_ok = (outer_len % 3 != 0) and (n_ip2 == 5 and n_ip4 == 5)
                        if not outer_ok or len(grand_examples) < 10:
                            grand_examples.append({
                                'n_G': n, 'tri_idx': tri_idx, 'v': v, 'i': i,
                                'cyclic_degs': cyc,
                                'n_i': n_i, 'n_ip1': n_ip1,
                                'n_ip2': n_ip2, 'n_ip3': n_ip3, 'n_ip4': n_ip4,
                                'outer_len': outer_len,
                                'outer_ok': outer_ok,
                                'graph6': G.canonical_label().graph6_string(),
                            })
        elapsed = time.time() - start
        print(f"n={n}: {pairs_n} (G,v) pairs, {triples_n} (G,v,i) triples; "
              f"{bad_triples_n} triples with both flank-adj deg ≥ 7 "
              f"[{elapsed:.0f}s]")
        sys.stdout.flush()
        grand_pairs += pairs_n
        grand_triples += triples_n
        grand_bad_triples += bad_triples_n
        grand_all_ge_7 += all_ge_7_n

    print()
    print("=" * 70)
    print(f"Grand totals across n_G in [12, {max_n}]:")
    print(f"  (G, v) pairs:                 {grand_pairs}")
    print(f"  (G, v, i) triples:            {grand_triples}")
    pct = 100 * grand_all_ge_7 / max(grand_pairs, 1)
    print(f"  pairs with all 5 neighbours deg ≥ 7: "
          f"{grand_all_ge_7} ({pct:.2f}%)")
    pct = 100 * grand_bad_triples / max(grand_triples, 1)
    print(f"  triples with BOTH flank-adj deg ≥ 7: "
          f"{grand_bad_triples} ({pct:.2f}%)")
    print()
    print("  Distribution of min(all 5 neighbour degs) per (G, v):")
    for m in sorted(grand_dist_min):
        c = grand_dist_min[m]
        ppct = 100 * c / max(grand_pairs, 1)
        bar = '#' * max(1, int(ppct))
        print(f"    {m}: {c:>6} ({ppct:5.2f}%) {bar[:50]}")
    print()
    print("  Distribution of MAX over i of "
          "min(n_i, n_{i+1}) per (G, v)\n"
          "  (i.e., the worst-case consecutive-pair-min --- if this is\n"
          "  ≤ 6, every i has min(n_i, n_{i+1}) ≤ 6 and the partial proof\n"
          "  applies; if it is ≥ 7, at least one i has both flanks ≥ 7):")
    for m in sorted(grand_dist_consec):
        c = grand_dist_consec[m]
        ppct = 100 * c / max(grand_pairs, 1)
        bar = '#' * max(1, int(ppct))
        print(f"    {m}: {c:>6} ({ppct:5.2f}%) {bar[:50]}")
    if grand_examples:
        print()
        n_outer_ok = sum(1 for ex in grand_examples if ex['outer_ok'])
        n_outer_bad = sum(1 for ex in grand_examples if not ex['outer_ok'])
        print(f"  All {len(grand_examples)} (G, v, i) triples with BOTH "
              f"flank-adj deg ≥ 7:")
        print(f"    {n_outer_ok} have n_{{i+2}}=n_{{i+4}}=5 "
              f"(F_outer candidate OK)")
        print(f"    {n_outer_bad} do NOT (F_outer might not apply)")
        print()
        for ex in grand_examples:
            tag = "[F_outer OK]" if ex['outer_ok'] else "[F_outer FAILS]"
            print(f"    {tag} n={ex['n_G']}, tri#{ex['tri_idx']}, "
                  f"v={ex['v']}, i={ex['i']}: "
                  f"cyc={ex['cyclic_degs']}, "
                  f"(n_i,n_{{i+1}},n_{{i+2}},n_{{i+3}},n_{{i+4}}) = "
                  f"({ex['n_i']},{ex['n_ip1']},{ex['n_ip2']},"
                  f"{ex['n_ip3']},{ex['n_ip4']}), "
                  f"F_outer len = {ex['outer_len']}")


if __name__ == '__main__':
    main()