experiments/check_c28_no_chord_apex_kempe_constancy.py iterates all
3 triangulations on 16 vertices with min degree 5 (whose duals are
the 28-vertex cubic plane graphs with face length ≥ 5 -- including
the C28 fullerene that disproves Conjecture 5.5). For each:
- applies every chord-apex reduction (every pentagonal face of the
dual × every rotation index i ∈ {0,…,4}),
- enumerates every proper 3-edge-colouring of each reduced dual,
- filters to chord-apex+Kempe colourings (Lemmas 5.X chord-apex +
Kempe-spike),
- traces K_b, K_c through the merged edge,
- computes h_φ via the CW rotation at each vertex,
- reports any colouring where h_φ is constant on V(K_b), V(K_c), or
both.
Result:
reductions tried : 60 + 60 + 70 = 190
chord-apex+Kempe colourings: 432 + 432 + 0 = 864
constant on V(K_b) : 0 + 0 + 0 = 0
constant on V(K_c) : 0 + 0 + 0 = 0
constant on both : 0 + 0 + 0 = 0
So even though the C28 fullerene admits a proper 3-edge-colouring on
which two intersecting Kempe cycles are both constant h_φ (the
Conjecture 5.5 counterexample), none of its chord-apex reductions
admits a chord-apex+Kempe colouring with the same property -- the
extra constraints (merged + spike same colour; K_b ⊇ {spike, side_0,
merged}; K_c ⊇ {spike, side_1, merged}) genuinely rule it out.
This is consistent with the broader empirical near-proof
(check_constancy_obstruction.py: 0/142,812 colourings constant) and
shows that the C28 obstruction-killing is not a fluke specific to
some smaller class; it works for the full chord-apex+Kempe layer.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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