coloring_nested_tire_graphs: empirical test of rainbow + König-lift on cut tires
For each cut tire on G'_1 of Holton-McKay #0 (HM cut: |S|=10,
matching 6-cut), brute-force enumerate proper edge 3-colorings,
compute the joint (σ_out, σ_in) projection, and check S_3-closure
and orbit decomposition.
Results (8 cut tires analyzed, 2 too big or trivial):
d face |f| out in |E| #col |π| S3-cl orbit sizes
1 0 12 5 0 17 96 93 yes [3, 6^15]
1 1 4 1 0 5 6 3 yes [3]
2 0 7 4 3 14 126 126 yes [6^21]
2 1 7 4 3 14 126 126 yes [6^21]
3 0-2 2 0 0 2 3 1 yes [1]
4 0 4 1 0 5 6 3 yes [3]
4 1 8 2 1 11 24 21 yes [3, 6^3]
5 1 2 0 0 2 3 1 yes [1]
6 0 12 3 2 17 96 93 yes [3, 6^15]
7 0 2 0 0 2 3 1 yes [1]
Findings:
1. S_3-closure is universal (structural, expected).
2. Orbit sizes are always 3 (constant) or 6 (generic).
3. Non-trivial cut tires have rich projections (e.g. d=2 has
21 size-6 orbits = 126 elements; d=6 has 16 orbits).
Neither conjecture is DIRECTLY testable on this example:
- Rainbow conjecture requires antipodal-chord SP face boundary
structure. Our cut tires' face boundaries don't naturally have
this shape.
- König-lift conjecture requires both sides give γ-face partitions
on a shared γ. Cut tires at consecutive depths share data via
in-spoke ↔ face-boundary-edge bijection, not via γ-face
partitions.
What CAN be observed: cut tire projections are LARGE and S_3-
symmetric (substantially looser than the rainbow case's 36-element
prediction). A "loose conjecture" would say π(T) ≥ c · 6 with c
depending only on |E(T)|, derivable from Prop 1.13 in paper.tex.
Files:
experiments/cut_tire_test.py
notes/cut_tire_conjecture_tests.tex (3 pages)
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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