experiments/check_v_neighbour_degrees.py reports the cyclic degree
sequence of v's 5 neighbours in G across all (G, v, i) triples
underlying chord-apex+Kempe colourings (|V(G)| ≤ 20).
Result:
- 100.00% of (G, v) pairs have at least one neighbour of v of
degree 5. So the partial proof of the deciding-face conjecture
(Theorem deciding-face-partial, which requires n_k ∈ {5, 6} for
some k) handles all (G, v) pairs IF we can pick any k freely.
- 99.70% of (G, v, i) triples have min(n_i, n_{i+1}) ≤ 6, so the
partial proof's flank-face argument applies to the specific i.
- 24 / 7,930 (0.30%) triples have BOTH n_i ≥ 7 AND n_{i+1} ≥ 7
(the "bad" case where the partial proof's flank-face doesn't
work). These occur at n_G = 19, 20 for triangulations with
cyclic neighbour-degree sequences like (5,7,7,5,5).
For these 24 bad triples, the remaining 3 neighbours typically have
degree 5, so F^♭_outer (length n_{i+2} + n_{i+4} - 3 = 5+5-3 = 7,
not divisible by 3) becomes the candidate deciding face. Extending
the structural proof to cover F^♭_outer is the next step.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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