Adds Proposition 1.13 (Edge-vertex coloring bijection for D(T)): for
a tire graph T satisfying the spoke-only hypothesis of Prop 1.8 (so
D(T) ~= C_{n+m} ∘ K_1), the number of proper 3-edge-colorings of D(T)
equals the number of proper 3-vertex-colorings of its interior dual
subgraph Γ ~= C_{n+m}, and both equal 2^{n+m} + 2 · (-1)^{n+m}.
Proof: Two bijection steps.
Step 1: Restriction is a bijection between proper 3-edge-colorings
of D(T) and proper 3-edge-colorings of the cycle C_L (where
L = n+m), because at each d_f the leaf's color is forced to be
the unique third color absent from the two cycle edges, and
leaves impose no further constraint.
Step 2: Proper 3-edge-colorings of C_L = proper 3-vertex-colorings
of L(C_L) = proper 3-vertex-colorings of C_L (since L(C_L) ~= C_L).
Step 3: Chromatic polynomial of C_L at k=3 is 2^L + 2 · (-1)^L.
Adds Remark 1.14 noting the closed form depends only on n+m, not
on the specific spoke-only annular triangulation or chord structure
of O.
Empirically verified for L in [3, 10] via Sage's chromatic
polynomials: edge-3-colorings of D(T) = vertex-3-colorings of C_L
= formula in every case.
Paper grows from 7 to 8 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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