diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.aux b/papers/coloring_nested_tire_graphs/notes/menagerie.aux index 4089a83..5dd1744 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.aux +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.aux @@ -1,2 +1,3 @@ \relax +\@writefile{toc}{\contentsline {paragraph}{Closed form.}{3}{}\protected@file@percent } \gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.log b/papers/coloring_nested_tire_graphs/notes/menagerie.log index 1a1cce1..98c8d48 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 21:05 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:18 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -293,40 +293,45 @@ Package pdftex.def Info: fig_corona.png used on input line 76. 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PDF statistics: - 95 PDF objects out of 1000 (max. 8388607) - 50 compressed objects within 1 object stream + 111 PDF objects out of 1000 (max. 8388607) + 60 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 31 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 df64440..549fe8e 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 7b37995..f9482e4 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.tex +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.tex @@ -101,20 +101,56 @@ processed edges incident to the new endpoint of $e$ (between $1$ and $2$, since $\Delta\le 3$). In practice this gives a clean product depending only on the degree sequence of $T$. -\subsection*{6.~Two-connected outerplanar with $\Delta=3$ is just $C_n$} +\subsection*{6.~Two-connected outerplanar with $\Delta\le 3$: cycle, possibly with a matching of chords} -The only $2$-connected outerplanar graphs are polygons (with optional chords). -Each chord adds a degree to each of its two endpoints; if every vertex on the -polygon $C_n$ already has degree $2$ from the cycle, then we may add at most -one chord-endpoint per vertex, so chords must form a matching. Already a -\emph{single} chord on a $2$-connected outerplanar graph forces both -endpoints to degree $3$. In particular, the only $2$-connected outerplanar -graph with $\Delta\le 3$ in which the maximum is actually attained at -\emph{every} vertex would be a polygon with a perfect matching of chords; -but each chord crosses some other (mod the planar embedding) unless the two -matched vertices are adjacent on the polygon, which collapses the -``$2$-connected'' assumption. The upshot is: \emph{the $2$-connected blocks -in our class are just cycles.} +The $2$-connected outerplanar graphs are polygons (with optional chords). +Each chord raises the degree of its two endpoints by $1$, and a polygon +vertex already has degree $2$ from the cycle. So for $\Delta\le 3$ the +chords must form a \emph{matching} of polygon vertices --- no vertex +can be an endpoint of two chords. In particular the simplest non-cycle +case is a polygon with a \emph{single} chord, denoted $\theta(1,p,q)$: +two paths of lengths $p$ and $q$ between two trivalent vertices, plus a +direct edge between those two trivalent vertices. Equivalently it is +$C_{p+q}$ with a chord joining the two cycle vertices at distance $p$ +apart on the polygon. + +\begin{center} +\textit{[$\theta(1,p,q)$: polygon $C_{p+q}$ with one chord; both chord +endpoints have degree~$3$, other polygon vertices have degree~$2$.]} +\end{center} + +$\theta(1,p,q)$ is outerplanar (the chord lies inside the polygon, and +all polygon vertices are on the outer face) and subcubic. + +\paragraph{Closed form.} +For each choice of chord color $c\in\{1,2,3\}$, the two cycle edges +incident to each chord endpoint must lie in $\{x,y\} = \{1,2,3\}\setminus\{c\}$. +Conditioning on whether the two pairs of cycle-edge colors at the chord +endpoints agree or disagree, and using the transfer matrix +$T = J - I$ on $K_3$ (eigenvalues $2$ and~$-1$ with multiplicities $1$ and $2$), +one finds +\[ +\boxed{\;P_e(\theta(1,p,q),\,3) +\;=\; \dfrac{2^{p+q} - 2^{p}(-1)^{q} - 2^{q}(-1)^{p} + 10\,(-1)^{p+q}}{3}\;}. +\] +Sanity checks: +\begin{itemize} +\item $p = q = 2$: $\theta(1,2,2) = K_4 - e$, formula gives + $(16 - 4 - 4 + 10)/3 = 6 = P_e(K_4 - e, 3)$. +\item $p = q = 3$: $\theta(1,3,3)$, formula gives + $(64 + 8 + 8 + 10)/3 = 30$. +\item $p = q = 6$: $\theta(1,6,6)$ (= the interior dual subgraph of + the partial tire dual for the barbell-$O$ tire of Figure~4 of + the main paper), formula gives $(4096 - 64 - 64 + 10)/3 = 1326$. +\end{itemize} +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. \subsection*{7.~Block--cut decomposition} @@ -140,20 +176,23 @@ cut-vertex constraint. For each cycle-block $B = C_n$ contributing $2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing $3$, this product is computable in time linear in $|V(G)|+|E(G)|$. -\section*{Outside the menagerie: theta graphs} +\section*{Outside the menagerie: $\theta(p,q,r)$ with all paths +$\ge 2$, i.e.\ $K_{2,3}$ subdivisions} \begin{center} \includegraphics[width=0.32\textwidth]{fig_theta.png} \end{center} -The complete bipartite graph $K_{2,3}$, equivalently the theta graph -$\theta(2,2,2)$, is \emph{not} outerplanar (it is a forbidden minor for -outerplanarity). In our tire-graph application this is the structure of the -interior dual subgraph of $D(T)$ when the inner outerplanar graph $O$ has a -bridge: two trivalent vertices $d_f$ connected by three internally -vertex-disjoint paths. Such a $D(T)$ falls outside the simple block-cut -menagerie above and its $P_e(\cdot,3)$ does not reduce to a product over -cycle-blocks; instead it is computed directly by deletion--contraction on the -theta-graph structure, or via a transfer matrix on the three paths. +The genuine theta graph $\theta(2,2,2) = K_{2,3}$ (and more generally any +$\theta(p,q,r)$ with $p,q,r\ge 2$) is \emph{not} outerplanar: $K_{2,3}$ +is a forbidden minor for outerplanarity, and every $\theta(p,q,r)$ with +$\min(p,q,r)\ge 2$ contains $K_{2,3}$ as a topological minor. Such graphs +do \emph{not} arise as the interior dual subgraph of a partial tire dual +$D(T)$ in this paper: when the inner outerplanar graph $O$ has a bridge, +the interior dual subgraph is the polygon-with-one-chord case +$\theta(1, p, q)$ of \S6, where the path of length~$1$ is precisely the +dual of the bridge. In particular $D(T)$ \emph{is} subcubic outerplanar +in the bridge case, and its edge $3$-coloring count is the +$\theta(1,p,q)$ closed form above. \end{document}