NEW Theorem 1.15: The tire treads from a single-vertex level
source S = {v_0} form a rooted tree T(G, S) under face containment.
Statement:
- Root: the depth-0 tire tread T_0 with degenerate outer
boundary {v_0} (the apex tire, B_out = {v_0}).
- Parent: for any tire tread T_c at depth d ≥ 1, the unique
parent T_p at depth d-1 is the tire whose inner outerplanar
graph O^(p) has B_out^(c) as one of its bounded faces.
Equivalently, R_c lies inside this bounded face of O^(p).
- Children: bijection with bounded faces of O^(p) whose
interior contains depth-≥(d+2) vertices.
Proof structure:
1. Root well-defined: G'_0 is connected (fan around v_0), so
unique component → unique T_0.
2. Existence of parent: faces immediately outside B_out^(c) on
the S-side have depth d-1, lie in some component of G'_{d-1}.
3. Uniqueness: by Proposition 1.7 (source-side simple-cycle
property), B_out^(c) is a simple cycle, and the depth-(d-1)
faces around it form a single contiguous arc in the dual,
hence one component → unique parent.
4. Children description: bounded faces of O^(p) are in bijection
with deeper component-tires.
5. Tree property: parent map strictly decreases depth, hence
no cycles, hence rooted tree.
Plus two clarifying remarks:
- Remark 1.16: multiple children iff O^(p) has multiple bounded
faces with non-trivial interiors. Spoke-only case → exactly
one child.
- Remark 1.17: combined with Theorem 1.9 (partition) and
Theorem 1.12 (outerplanar inner dual), any coloring problem
on G factors through:
• local outerplanar coloring on each tread,
• parent-child consistency along shared B_out^(c) cycles.
This is the structural setup for the chain-pigeonhole program.
Page count: 10 → 11.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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