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. (pdftex.def) Requested size: 258.36668pt x 157.34296pt. -[4 <./fig_3chord_example.png>] [5 <./fig_blocktree.png>] (./menagerie.aux) ) + [5 <./fig_blocktree.png>] (./menagerie.aux) ) Here is how much of TeX's memory you used: 4641 strings out of 478268 75554 string characters out of 5846347 - 373157 words of memory out of 5000000 + 374157 words of memory out of 5000000 22818 multiletter control sequences out of 15000+600000 479481 words of font info for 67 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 55i,8n,63p,245b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s -{/u -sr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc} -Output written on menagerie.pdf (5 pages, 321355 bytes). + 55i,10n,63p,245b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s +{/usr/local/texlive/2022/texmf-di +st/fonts/enc/dvips/cm-super/cm-super-ts1.enc} +Output written on menagerie.pdf (5 pages, 335002 bytes). PDF statistics: - 116 PDF objects out of 1000 (max. 8388607) - 62 compressed objects within 1 object stream + 121 PDF objects out of 1000 (max. 8388607) + 65 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 36 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.pdf b/papers/coloring_nested_tire_graphs/notes/menagerie.pdf index 069db0d..629c4da 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/menagerie.pdf and b/papers/coloring_nested_tire_graphs/notes/menagerie.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.tex b/papers/coloring_nested_tire_graphs/notes/menagerie.tex index bbc1203..42fce8a 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.tex +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.tex @@ -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