From e29836c78ace3346dbd2d91781d265e2a3d648ac Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 21:53:59 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20add=20gene?= =?UTF-8?q?ral=20chromatic-polynomial=20method=20for=20=C2=A76=20graphs?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 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 --- .../notes/menagerie.aux | 1 + .../notes/menagerie.log | 68 ++++++++++-------- .../notes/menagerie.pdf | Bin 321355 -> 335002 bytes .../notes/menagerie.tex | 39 ++++++++-- 4 files changed, 75 insertions(+), 33 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.aux b/papers/coloring_nested_tire_graphs/notes/menagerie.aux index 3242b1d..10acf49 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.aux +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.aux @@ -1,4 +1,5 @@ \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 } \gdef \@abspage@last{5} diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.log b/papers/coloring_nested_tire_graphs/notes/menagerie.log index 0110919..d166b20 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.log +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:30 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:53 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -296,47 +296,57 @@ File: fig_theta133.png Graphic file (type png) Package pdftex.def Info: fig_theta133.png used on input line 118. (pdftex.def) Requested size: 272.45877pt x 107.18588pt. [3 <./fig_theta133.png>] - +Overfull \hbox (16.04408pt too wide) in paragraph at lines 168--176 +\OT1/cmr/m/n/10.95 a transfer-matrix prod-uct on the poly-gon: edge $\OML/cmm/m +/it/10.95 e[]$\OT1/cmr/m/n/10.95 's al-lowed colour set is $[\OML/cmm/m/it/10.9 +5 k\OT1/cmr/m/n/10.95 ] \OMS/cmsy/m/n/10.95 n f\OML/cmm/m/it/10.95 c[] \OT1/cmr +/m/n/10.95 : \OML/cmm/m/it/10.95 v[][]v[][]j\OMS/cmsy/m/n/10.95 g$\OT1/cmr/m/n/ +10.95 , + [] + + File: fig_3chord_example.png Graphic file (type png) -Package pdftex.def Info: fig_3chord_example.png used on input line 161. +Package pdftex.def Info: fig_3chord_example.png used on input line 192. (pdftex.def) Requested size: 328.82707pt x 160.37921pt. - +[4 <./fig_3chord_example.png>] + File: fig_blocktree.png Graphic file (type png) -Package pdftex.def Info: fig_blocktree.png used on input line 211. +Package pdftex.def Info: fig_blocktree.png used on input line 242. 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