Per user spec: instead of including the actual depth-(d±1) edges
incident to the face boundary, redefine the cut tire as:
- Face boundary walk of f (depth-d edges in H_d).
- For each vertex v on the boundary walk with degree-2 in H_d:
add a fresh vertex n_v and fresh edge {v, n_v}, labelled
"out spoke" if v has an incident depth-(d-1) edge in G'_i,
"in spoke" if v has an incident depth-(d+1) edge.
Result: each cut tire is intrinsically "cycle (or closed walk) +
labelled pendants," structurally isomorphic to the partial tire
dual D(T) from paper.tex. Pendants ↔ D(T)'s leaves, face boundary
↔ T'_ann.
This means propositions about D(T) (chromatic polynomial counts,
S_3-orbit structure, rainbow conjecture, etc.) apply verbatim to
each cut tire.
Updates:
- notes/cut_depth_label.tex: Definition rewritten, structural
remark added, table of spoke counts updated to match new defn.
- experiments/cut_tire.py: cut_tire_at() now computes labelled
pendants instead of incident edges; draw_cut_tire renders
pendant vertices (orange squares for out, green squares for in)
with edges offset toward parent-graph neighbor.
- notes/fig_cut_tire.png: regenerated.
Note grows to 6 pages.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
cb6a79f799