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coloring_nested_tire_graphs: fix menagerie -- θ(1,p,q) IS outerplanar (cycle + chord)

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User correctly pointed out:
(1) The Figure 4 partial-tire-dual interior structure is not a "theta
    graph" in the K_{2,3} sense (which requires all three paths of
    length ≥ 2).  It is θ(1, 6, 6): a 12-cycle with one chord.
(2) θ(1, p, q) IS outerplanar (just a polygon with one chord), so it
    belongs IN the menagerie, not outside it.

Revisions:

- Section 6 ("2-connected outerplanar with Δ ≤ 3"): previously claimed
  the class is just cycles; corrected to "cycle, possibly with a
  matching of chords."  Added explicit description of θ(1, p, q) and
  a closed-form for its proper 3-edge-coloring count:

    P_e(θ(1,p,q), 3) = (2^{p+q} - 2^p (-1)^q - 2^q (-1)^p + 10 (-1)^{p+q}) / 3.

  Verified against Sage's chromatic polynomial for all p, q ∈ {2..6}.

- "Outside the menagerie" section: previously said "theta graphs (all
  flavours) are not outerplanar."  Corrected to clarify that only
  θ(p, q, r) with all three paths of length ≥ 2 (= K_{2,3} subdivisions)
  is not outerplanar.  Explicitly noted that the bridge-case partial
  tire dual gives θ(1, p, q) which IS in the menagerie, with edge-3-
  coloring count given by the closed form.

The Figure 4 partial-tire-dual (m=4 outer cycle + barbell O with
bridge) has θ(1, 6, 6) as its interior dual subgraph and so admits
exactly 1326 proper 3-edge-colorings on the interior cycle-with-
chord; leaves contribute their forced colors as in the spoke-only
case.

Paper unchanged.  This is a correction within the notes/ subdir only.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-25 21:19:08 -04:00
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papers/coloring_nested_tire_graphs/notes/menagerie.aux
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papers/coloring_nested_tire_graphs/notes/menagerie.tex
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@@ -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 $\Delta\le 3$). In practice this gives a clean product depending only on the
degree sequence of $T$. 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). The $2$-connected outerplanar graphs are polygons (with optional chords).
Each chord adds a degree to each of its two endpoints; if every vertex on the Each chord raises the degree of its two endpoints by $1$, and a polygon
polygon $C_n$ already has degree $2$ from the cycle, then we may add at most vertex already has degree $2$ from the cycle. So for $\Delta\le 3$ the
one chord-endpoint per vertex, so chords must form a matching. Already a chords must form a \emph{matching} of polygon vertices --- no vertex
\emph{single} chord on a $2$-connected outerplanar graph forces both can be an endpoint of two chords. In particular the simplest non-cycle
endpoints to degree $3$. In particular, the only $2$-connected outerplanar case is a polygon with a \emph{single} chord, denoted $\theta(1,p,q)$:
graph with $\Delta\le 3$ in which the maximum is actually attained at two paths of lengths $p$ and $q$ between two trivalent vertices, plus a
\emph{every} vertex would be a polygon with a perfect matching of chords; direct edge between those two trivalent vertices. Equivalently it is
but each chord crosses some other (mod the planar embedding) unless the two $C_{p+q}$ with a chord joining the two cycle vertices at distance $p$
matched vertices are adjacent on the polygon, which collapses the apart on the polygon.
``$2$-connected'' assumption. The upshot is: \emph{the $2$-connected blocks
in our class are just cycles.} \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} \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 $2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing
$3$, this product is computable in time linear in $|V(G)|+|E(G)|$. $3$, 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} \begin{center}
\includegraphics[width=0.32\textwidth]{fig_theta.png} \includegraphics[width=0.32\textwidth]{fig_theta.png}
\end{center} \end{center}
The complete bipartite graph $K_{2,3}$, equivalently the theta graph The genuine theta graph $\theta(2,2,2) = K_{2,3}$ (and more generally any
$\theta(2,2,2)$, is \emph{not} outerplanar (it is a forbidden minor for $\theta(p,q,r)$ with $p,q,r\ge 2$) is \emph{not} outerplanar: $K_{2,3}$
outerplanarity). In our tire-graph application this is the structure of the is a forbidden minor for outerplanarity, and every $\theta(p,q,r)$ with
interior dual subgraph of $D(T)$ when the inner outerplanar graph $O$ has a $\min(p,q,r)\ge 2$ contains $K_{2,3}$ as a topological minor. Such graphs
bridge: two trivalent vertices $d_f$ connected by three internally do \emph{not} arise as the interior dual subgraph of a partial tire dual
vertex-disjoint paths. Such a $D(T)$ falls outside the simple block-cut $D(T)$ in this paper: when the inner outerplanar graph $O$ has a bridge,
menagerie above and its $P_e(\cdot,3)$ does not reduce to a product over the interior dual subgraph is the polygon-with-one-chord case
cycle-blocks; instead it is computed directly by deletion--contraction on the $\theta(1, p, q)$ of \S6, where the path of length~$1$ is precisely the
theta-graph structure, or via a transfer matrix on the three paths. 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} \end{document}