math-research/papers/face_monochromatic_pairs/experiments/verify_28_vertex_counterexample.py

"""Verify the n=28 counterexample (smallest dual of min-degree-5
triangulation that violates the face-length-≥-5 form of Conjecture
5.5) and render a planar PNG of it.

The graph H is the planar dual of a 16-vertex triangulation with
min degree ≥ 5 (the 3rd one in sage's enumeration). It has 28
vertices, 42 edges, and all faces of length ≥ 5. The colouring shown
below makes K_{red, blue} and K_{red, green} both 12-cycles sharing
the colour-red edge (0, 1) with h_φ ≡ -1 on each.

Run with: sage experiments/verify_28_vertex_counterexample.py
"""
import os
import sys
import math

from sage.all import Graph

HERE = os.path.dirname(os.path.abspath(__file__))
sys.path.insert(0, HERE)
from search_smaller_counterexample import (
    edge_key, heawood_numbers, trace_kempe,
)

OUT_PNG = os.path.join(HERE, '..', 'figures', 'min-face-5-counterexample.png')

# Dual edges and the discovered colouring (sorted-edge order).
EDGES_LIST = [
    (0, 1), (0, 4), (0, 6), (1, 2), (1, 5), (2, 3), (2, 8), (3, 4),
    (3, 11), (4, 13), (5, 7), (5, 9), (6, 7), (6, 15), (7, 17),
    (8, 10), (8, 12), (9, 10), (9, 19), (10, 20), (11, 12), (11, 14),
    (12, 22), (13, 14), (13, 16), (14, 23), (15, 16), (15, 18),
    (16, 25), (17, 18), (17, 19), (18, 26), (19, 21), (20, 21),
    (20, 22), (21, 27), (22, 24), (23, 24), (23, 25), (24, 27),
    (25, 26), (26, 27),
]
COLOURING = (0, 1, 2, 2, 1, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 2,
             1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 2, 1, 1,
             2, 1, 2, 0, 1, 2)
COLOUR_NAME = {0: 'red', 1: 'blue', 2: 'green'}


def build():
    G = Graph(multiedges=False, loops=False)
    for u, v in EDGES_LIST:
        G.add_edge(u, v)
    # Sage's sorted edges should align with EDGES_LIST.
    edges_sorted = sorted([edge_key(u, v) for (u, v) in G.edge_iterator(labels=False)])
    assert edges_sorted == EDGES_LIST, "edge order mismatch"
    col_of_edge = {edges_sorted[i]: COLOURING[i] for i in range(len(edges_sorted))}
    return G, col_of_edge


def main():
    G, col_of_edge = build()
    print(f"|V| = {G.order()}, |E| = {G.size()}")
    assert G.is_planar(set_embedding=True)
    print(f"face lengths: {sorted([len(f) for f in G.faces()])}")
    assert all(len(f) >= 5 for f in G.faces()), "face length < 5"
    # Verify proper 3-edge-colouring.
    for v in G.vertex_iterator():
        cs = sorted(col_of_edge[edge_key(v, u)] for u in G.neighbors(v))
        assert cs == [0, 1, 2], f"vertex {v} colours {cs}"
    print("proper 3-edge-colouring ✓")

    h = heawood_numbers(G, col_of_edge)
    plus = sum(1 for v in h if h[v] == +1)
    minus = sum(1 for v in h if h[v] == -1)
    print(f"global h_φ: {plus} (+1) / {minus} (-1)")

    # Verify both Kempe cycles through (0, 1) are constant.
    K0 = trace_kempe(G, col_of_edge, (0, 1), (0, 1))   # red+blue
    K1 = trace_kempe(G, col_of_edge, (0, 1), (0, 2))   # red+green
    h_K0 = [h[v] for v in K0]
    h_K1 = [h[v] for v in K1]
    print(f"K_{{red, blue}}  (len {len(K0)}): {K0}, h = {h_K0[0]} (const? {len(set(h_K0))==1})")
    print(f"K_{{red, green}} (len {len(K1)}): {K1}, h = {h_K1[0]} (const? {len(set(h_K1))==1})")
    print(f"V(K0) ∪ V(K1)| = {len(set(K0) | set(K1))} / {G.order()}")

    # Canonical graph6
    print(f"canonical graph6 = {G.canonical_label().graph6_string()}")

    # Render PNG with planar layout
    import matplotlib
    matplotlib.use('Agg')
    import matplotlib.pyplot as plt

    pos = G.layout_planar()
    fig, ax = plt.subplots(figsize=(10, 10), dpi=160)
    ax.set_aspect('equal')
    ax.axis('off')
    for (u, v) in G.edge_iterator(labels=False):
        c = COLOUR_NAME[col_of_edge[edge_key(u, v)]]
        x1, y1 = pos[u]; x2, y2 = pos[v]
        ax.plot([x1, x2], [y1, y2], color=c, linewidth=2.0, solid_capstyle='round')
    K0set = set(K0); K1set = set(K1)
    for v, (x, y) in pos.items():
        if v in K0set and v in K1set:
            face_c = '#bbbbff'   # on both cycles
        elif v in K0set or v in K1set:
            face_c = '#ddddff'
        else:
            face_c = 'lightgrey'
        ax.plot(x, y, 'o', markersize=18, markerfacecolor=face_c,
                markeredgecolor='black', markeredgewidth=1.0)
        ax.text(x, y, str(v), ha='center', va='center', fontsize=8)
    xs = [p[0] for p in pos.values()]
    ys = [p[1] for p in pos.values()]
    pad = 0.5
    ax.set_xlim(min(xs) - pad, max(xs) + pad)
    ax.set_ylim(min(ys) - pad, max(ys) + pad)
    os.makedirs(os.path.dirname(OUT_PNG), exist_ok=True)
    fig.tight_layout()
    fig.savefig(OUT_PNG, dpi=160, bbox_inches='tight')
    plt.close(fig)
    print(f"\nWrote: {OUT_PNG}")


if __name__ == '__main__':
    main()