NEW: Theorem 1.5 (Tire treads partition the bounded faces).
For a maximal planar graph G with level source S on the outer face,
the family of tire treads { R_{C'} : d ≥ 0, C' a connected component
of G'_d } supplied by the tire-component lemma partitions the
bounded part of |Π_G|:
(i) every bounded face of G lies in exactly one tread R_{C'};
(ii) distinct treads have disjoint interiors.
Proof: each bounded face has a unique dual depth d, hence its dual
vertex lies in G'_d alone, and within G'_d in a unique component C'.
By the tire-component lemma, that C' carries the unique tread
containing the face.
This is the first step toward a chain pigeonhole argument that
colorings extend across the nested tire treads induced by a level
source.
Paper grows to 10 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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