face_monochromatic_pairs: strengthen Conj 5.5 to face-length ≥ 5, find 28-vertex counterexample
- Paper: Conjecture 5.5 restated with the hypothesis that every face
of H has length ≥ 5 (= the cubic plane analogue of "no triangles
or quadrilaterals as faces"). This kills K_4 and the n=8 trivial
counterexamples (girth 3) and the ad-hoc n=40 counterexample
(which has 2 triangles and 4 quadrilaterals). A new remark catalogues
these excluded counterexamples and the smallest cubic plane graphs
satisfying the hypothesis (dodecahedron at |V| = 20).
- search_smaller_counterexample.py: --min-face=N option to filter
cubic planar graphs by minimum face length.
- search_min_face5_counterexample.py: enumerates triangulations T
with min degree ≥ 5 via graphs.triangulations(n, minimum_degree=5),
takes planar dual (= cubic plane with all faces ≥ 5), and runs the
Heawood-constancy check.
- Result: smallest counterexample at triangulation order n_T = 16,
whose dual is a 28-vertex cubic plane graph (graph6
[kG[A?_A?_?_?K?D?@_CO?o?@_??A??@C??O??AG?C????`???a???W???A_???F).
Faces: 12 pentagons + 4 hexagons (a C28 fullerene). Both
K_{red, blue} and K_{red, green} are 12-cycles sharing the
colour-red edge (0, 1) and both have h_φ ≡ -1. 8 of 28 vertices
lie outside V(K_0) ∪ V(K_1).
- verify_28_vertex_counterexample.py: reproduces the counterexample,
verifies all properties, and renders figures/min-face-5-counterexample.png.
Note on the boundary: face-length ≥ 6 is impossible for cubic plane
graphs by Euler (6F = 6(V/2 + 2) > 3V = sum face lengths for V > 4).
So face-length ≥ 5 is the strongest face-length restriction admitting
any cubic plane graphs at all.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""Verify the n=28 counterexample (smallest dual of min-degree-5
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triangulation that violates the face-length-≥-5 form of Conjecture
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5.5) and render a planar PNG of it.
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The graph H is the planar dual of a 16-vertex triangulation with
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min degree ≥ 5 (the 3rd one in sage's enumeration). It has 28
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vertices, 42 edges, and all faces of length ≥ 5. The colouring shown
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below makes K_{red, blue} and K_{red, green} both 12-cycles sharing
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the colour-red edge (0, 1) with h_φ ≡ -1 on each.
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Run with: sage experiments/verify_28_vertex_counterexample.py
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"""
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import os
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import sys
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import math
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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 search_smaller_counterexample import (
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edge_key, heawood_numbers, trace_kempe,
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)
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OUT_PNG = os.path.join(HERE, '..', 'figures', 'min-face-5-counterexample.png')
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# Dual edges and the discovered colouring (sorted-edge order).
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EDGES_LIST = [
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(0, 1), (0, 4), (0, 6), (1, 2), (1, 5), (2, 3), (2, 8), (3, 4),
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(3, 11), (4, 13), (5, 7), (5, 9), (6, 7), (6, 15), (7, 17),
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(8, 10), (8, 12), (9, 10), (9, 19), (10, 20), (11, 12), (11, 14),
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(12, 22), (13, 14), (13, 16), (14, 23), (15, 16), (15, 18),
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(16, 25), (17, 18), (17, 19), (18, 26), (19, 21), (20, 21),
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(20, 22), (21, 27), (22, 24), (23, 24), (23, 25), (24, 27),
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(25, 26), (26, 27),
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]
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COLOURING = (0, 1, 2, 2, 1, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 1, 2, 2,
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1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 0, 2, 1, 1,
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2, 1, 2, 0, 1, 2)
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COLOUR_NAME = {0: 'red', 1: 'blue', 2: 'green'}
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def build():
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G = Graph(multiedges=False, loops=False)
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for u, v in EDGES_LIST:
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G.add_edge(u, v)
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# Sage's sorted edges should align with EDGES_LIST.
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edges_sorted = sorted([edge_key(u, v) for (u, v) in G.edge_iterator(labels=False)])
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assert edges_sorted == EDGES_LIST, "edge order mismatch"
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col_of_edge = {edges_sorted[i]: COLOURING[i] for i in range(len(edges_sorted))}
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return G, col_of_edge
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def main():
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G, col_of_edge = build()
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print(f"|V| = {G.order()}, |E| = {G.size()}")
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assert G.is_planar(set_embedding=True)
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print(f"face lengths: {sorted([len(f) for f in G.faces()])}")
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assert all(len(f) >= 5 for f in G.faces()), "face length < 5"
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# Verify proper 3-edge-colouring.
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for v in G.vertex_iterator():
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cs = sorted(col_of_edge[edge_key(v, u)] for u in G.neighbors(v))
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assert cs == [0, 1, 2], f"vertex {v} colours {cs}"
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print("proper 3-edge-colouring ✓")
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h = heawood_numbers(G, col_of_edge)
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plus = sum(1 for v in h if h[v] == +1)
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minus = sum(1 for v in h if h[v] == -1)
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print(f"global h_φ: {plus} (+1) / {minus} (-1)")
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# Verify both Kempe cycles through (0, 1) are constant.
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K0 = trace_kempe(G, col_of_edge, (0, 1), (0, 1)) # red+blue
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K1 = trace_kempe(G, col_of_edge, (0, 1), (0, 2)) # red+green
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h_K0 = [h[v] for v in K0]
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h_K1 = [h[v] for v in K1]
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print(f"K_{{red, blue}} (len {len(K0)}): {K0}, h = {h_K0[0]} (const? {len(set(h_K0))==1})")
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print(f"K_{{red, green}} (len {len(K1)}): {K1}, h = {h_K1[0]} (const? {len(set(h_K1))==1})")
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print(f"V(K0) ∪ V(K1)| = {len(set(K0) | set(K1))} / {G.order()}")
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# Canonical graph6
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print(f"canonical graph6 = {G.canonical_label().graph6_string()}")
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# Render PNG with planar layout
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import matplotlib
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matplotlib.use('Agg')
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import matplotlib.pyplot as plt
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pos = G.layout_planar()
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fig, ax = plt.subplots(figsize=(10, 10), dpi=160)
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ax.set_aspect('equal')
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ax.axis('off')
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for (u, v) in G.edge_iterator(labels=False):
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c = COLOUR_NAME[col_of_edge[edge_key(u, v)]]
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color=c, linewidth=2.0, solid_capstyle='round')
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K0set = set(K0); K1set = set(K1)
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for v, (x, y) in pos.items():
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if v in K0set and v in K1set:
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face_c = '#bbbbff' # on both cycles
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elif v in K0set or v in K1set:
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face_c = '#ddddff'
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else:
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face_c = 'lightgrey'
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ax.plot(x, y, 'o', markersize=18, markerfacecolor=face_c,
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markeredgecolor='black', markeredgewidth=1.0)
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ax.text(x, y, str(v), ha='center', va='center', fontsize=8)
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xs = [p[0] for p in pos.values()]
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ys = [p[1] for p in pos.values()]
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pad = 0.5
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ax.set_xlim(min(xs) - pad, max(xs) + pad)
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ax.set_ylim(min(ys) - pad, max(ys) + pad)
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os.makedirs(os.path.dirname(OUT_PNG), exist_ok=True)
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fig.tight_layout()
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fig.savefig(OUT_PNG, dpi=160, bbox_inches='tight')
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plt.close(fig)
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print(f"\nWrote: {OUT_PNG}")
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if __name__ == '__main__':
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main()
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