Added section "Could the minimum non-trivial cyclic cut be 8?"
Answer: yes in principle. Birkhoff gives "≥ 6"; nothing in the
condition pins the value to 6. A planar cubic G^* with min
non-facial cyclic edge connectivity = 8 would have:
- No non-facial 6-edge cut.
- No non-facial 7-edge cut.
- Some non-facial 8-edge cut.
By cut-parity lemma: 8-cuts have even sides; 7-cuts have odd
sides.
Heuristic when this happens: link vertices of degree ≥ 6
(rather than icosahedron-tight 5) push second-link length to
≥ 10, eliminating small non-facial separators.
The cut-tire framework adapts: chain DP and T_∂ construction
work for any cut size, with parameter changes. Per-tire half
needs re-examining for larger structures.
Bottom line: min non-trivial cyclic cut is one of {6, 7, 8, 9,
...}; Birkhoff doesn't pin it down. Framework's natural domain
is whatever value it happens to take, with 6 being the simplest.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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