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coloring_nested_tire_graphs: add general chromatic-polynomial method for §6 graphs

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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>
This commit is contained in:
didericis
2026-05-25 21:53:59 -04:00
parent dfca45e913
commit e29836c78a
4 changed files with 75 additions and 33 deletions
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papers/coloring_nested_tire_graphs/notes/menagerie.aux
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\relax
\@writefile{toc}{\contentsline {paragraph}{Closed form.}{3}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{General method.}{4}{}\protected@file@percent }
\@writefile{toc}{\contentsline {paragraph}{Example calculation (three chords on $C_8$).}{4}{}\protected@file@percent }
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papers/coloring_nested_tire_graphs/notes/menagerie.log
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\OT1/cmr/m/n/10.95 a transfer-matrix prod-uct on the poly-gon: edge $\OML/cmm/m
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papers/coloring_nested_tire_graphs/notes/menagerie.tex
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@@ -146,10 +146,41 @@ The formula has been verified empirically against \texttt{Sage}'s
chromatic polynomial routine for all
$p, q \in \{2, 3, 4, 5, 6\}$.
More generally, a polygon with $r$ chords forming a matching has
chromatic polynomial computable by the same transfer-matrix idea
along the $r{+}1$ paths between consecutive chord endpoints on the
polygon, with a product constraint at each chord endpoint.
\paragraph{General method.} For $G = C_n + M$ where $M$ is a
matching of non-crossing chords, the edge chromatic polynomial can be
computed by summing over chord-color assignments and applying a
constrained transfer matrix to the polygon edges:
\[
P_e(G, k) \;=\; \sum_{(c_1, \dots, c_r) \in [k]^r}\, N\bigl(C_n;\,
\text{forbidden}(c_1,\dots,c_r),\,k\bigr),
\]
where $N(C_n; F, k)$ counts proper $k$-edge-colorings of the polygon
$C_n$ subject to: for each cycle edge $e_i$, the colour of $e_i$
avoids every chord-colour $c_j$ such that an endpoint of chord $j$
is incident to $e_i$ (i.e.\ $v_i$ or $v_{i+1}$ is an endpoint of
chord $j$).
Concretely, with chord $j$ at polygon vertices $v_{a_j}, v_{b_j}$
and chord colour $c_j$, the constraints on cycle-edge colours are:
\begin{align*}
\text{at } v_{a_j}\text{: } &\quad c(e_{a_j-1}) \ne c_j,\quad c(e_{a_j}) \ne c_j,\\
\text{at } v_{b_j}\text{: } &\quad c(e_{b_j-1}) \ne c_j,\quad c(e_{b_j}) \ne c_j,
\end{align*}
on top of the usual cycle-adjacency $c(e_i) \ne c(e_{i+1})$. For
each chord-colour assignment, $N(C_n; F, k)$ is a transfer-matrix
product on the polygon: edge $e_i$'s allowed colour set is
$[k] \setminus \{c_j : v_i\text{ or }v_{i+1}\text{ on chord }j\}$,
and the proper-coloring transitions $c(e_i)\to c(e_{i+1})$ go through
a constrained $J - I$ matrix. The full sum is computed in time
polynomial in $n$ and $r$.
For $r = 1$ this reduces to the closed form for $\theta(1,p,q)$
above (sum over the single $c_1$, then transfer-matrix on the two
arcs). For larger $r$ there is no single closed form independent
of the chord placement, but the computation is mechanical. The
$3$-chord example below illustrates how chord-colour assignments
can be eliminated by direct constraint propagation when the cycle
is short enough.
\paragraph{Example calculation (three chords on $C_8$).}
Take the polygon $C_8$ with the three non-crossing chords