Adds a 'General method' paragraph to menagerie §6 describing how to
compute P_e(G, k) for any G = C_n + (matching of non-crossing chords):
P_e(G, k) = Σ_{(c_1,...,c_r)} N(C_n; forbidden(c_1,...,c_r), k)
where the sum is over chord-color assignments and N counts proper
k-edge-colorings of C_n subject to per-edge forbidden colors (= the
chord colors at adjacent chord endpoints). For each chord-color
choice the inner count is a transfer-matrix product on the polygon,
computed in O(n k^2) time, so the full polynomial is computable in
O(n k^{r+2}) time.
The method specializes to the closed form for θ(1, p, q) (r = 1) and
generalizes to any number of non-crossing chords. Verified against
Sage's chromatic polynomial of the line graph on:
- θ(1, 3, 3): 30
- C_8 + {(0,2), (3,7), (4,6)}: 6
- C_10 + {(0,2), (3,5), (6,8)}: 18
Note grows by ~1/2 page; still 5 pages total.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
e29836c78a