Previous version had loose formulas and overstated what second-link
length forces. Replaced with cleaner version that:
- States the maximal-planar constraints explicitly
(E = 3V-6, F = 2V-4, sum of deg = 6V-12).
- Notes the FORCED 12 degree-5 vertices when all degrees ∈ {5,6}.
- Gives the correct second-link length formula:
L_2(v) = d + sum_{u in link(v)} (deg(u) - 5)
Earlier version had this wrong.
- Concretely: pentakis dodecahedron has L_2 = 10 around every
vertex, but its dual (Buckyball) STILL has 6-edge cyclic cuts
arising from non-second-link constructions.
So second-link length being large doesn't prevent small non-facial
cyclic cuts via other separators. The min cut size is not pinned
down by local link structure alone.
Bottom line unchanged: min non-facial cyclic cut for a min 4CT
counterexample could be 6, 7, 8, ... and Birkhoff alone doesn't
distinguish.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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