NEW NOTE: birkhoff_internally_6_connected.tex (3 pages)
NEW SCRIPT: experiments/draw_internally_6_connected.py
NEW FIGURE: icosahedron_internally_6_connected.pdf
States and illustrates the Birkhoff (1913) condition that any
minimum 4CT counterexample must be internally 6-connected:
- No separating 3-cycle.
- No separating 4-cycle.
- No separating 5-cycle isolating ≥ 2 vertices on either side.
- Only separating 5-cycles isolating exactly 1 vertex.
The icosahedron is the canonical example: 12 vertices all of
degree 5; the 5 neighbors of every vertex form a 5-cycle whose
removal isolates that vertex. Sage verification confirms this:
Vertex 0 has 5 neighbors: [1, 5, 7, 8, 11]
Induced subgraph on neighbors: 5 edges, is_cycle=True
After removing the 5 neighbors: 2 components, sizes=[1, 6]
Note also lists the graphs used in our framework testing:
- Icosahedron (12 v, dual = dodecahedron)
- Pentakis dodecahedron (32 v, dual = Buckyball)
- Holton-McKay graphs (21 v primal, 38 v dual)
All are internally 6-connected, hence in the framework's intended
domain.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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