NEW Theorem 1.12: For any tire graph T, the inner dual Γ of its
tire tread (= subgraph of D(T) induced on interior dual vertices)
is outerplanar.
The theorem also gives a constructive characterization: Γ admits a
planar embedding as a (possibly non-simple) Hamilton walk through
every d_f, plus zero or more non-crossing chords.
Proof structure (constructive):
Case 1 (R is a disk, one boundary degenerate): the polygon
triangulation has no interior vertex, so its dual is a tree
(p-2 vertices, p-3 diagonals). Trees are outerplanar.
Case 2 (R is an annulus, both boundaries non-degenerate):
Step 1 - Cyclic ordering: cut R along any spoke e* to convert
the annulus into a closed disk. The disk boundary traverses
B_out + e* + B_in (reverse) + e*, yielding a cyclic sequence
S of annular faces with multiplicities (one per boundary edge,
+ detours for boundary-free faces).
Step 2 - Hamilton walk: consecutive entries of S share an
interior annular edge or coincide; the resulting closed walk
in Γ visits every d_f (using detours for the rare interior
annular triangles with zero boundary edges).
Step 3 - Non-crossing chords: remaining interior annular edges
become chords. Since the underlying E_ann edges in T are
non-crossing in the planar embedding, the chords are
non-crossing in Γ.
Step 4 - Outerplanar layout: place the |F_ann| vertices on a
circle in S-order, draw walk edges as the circle, chords inside.
All vertices on outer face → outerplanar.
Two remarks following:
Remark 1.13: spoke-only case is the classical Hamilton cycle
Γ ≅ C_{n+m} with zero chords.
Remark 1.14: bridge case (O with a bridge whose 2 incident faces
are annular) gives the theta graph Θ(1, b, c) — Hamilton cycle of
length n + m_∂ plus a single length-1 chord. The length-1 chord
contributes no degree-2 branch vertex to a K_{2,3} subdivision,
explaining why this is outerplanar despite being a theta graph.
Foundational paper grows from 7 to 8 pages.
This theorem unlocks the chain pigeonhole argument over tire
treads: each tread's coloring problem is on an outerplanar dual
graph, where the structure is locally tractable.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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