didericis
b526d204ff
UPDATED: birkhoff_internally_6_connected.tex now adds the distinction
between "internally 6-connected" (= cyclic edge conn ≥ 6 in dual)
and the framework's needed condition (= cyclic edge conn EXACTLY 6,
so 6-edge cuts exist). Notes that this is a real a priori
restriction not provided by Birkhoff alone.
NEW NOTE: even_separating_cycle.tex (3 pages)
Addresses: "must a min 4CT counterexample have a separating n-cycle
with n even and n ≥ 6?"
Honest answer: I don't know of a proof either way.
Key contributions:
- Lemma (cut-parity in cubic graphs): |C| ≡ |S| ≡ |T| (mod 2).
So even-length cycles in primal G ↔ cuts with even-sized sides
in dual G^*.
- |V(G^*)| = 2|V(G)| - 4 is always even, so both sides have
matching parity.
- Birkhoff doesn't rule out odd-length separating cycles ≥ 7.
- Second-link heuristic: in internally 6-conn triangulations,
the "second link" of any vertex is typically a 6-cycle, giving
abundant separating 6-cycles in practice. But this is
heuristic, not proven for all such triangulations.
Conjecture (stated, not proven): every internally 6-conn planar
triangulation with ≥ 12 vertices has a separating even n-cycle
with n ≥ 6.
Equivalent: every planar cubic graph with cyclic edge connectivity
≥ 6 and ≥ 20 vertices has a cyclic edge cut of size exactly 6.
This is a structural question; I don't know a planar cubic
counterexample.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>