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\title{Edge $3$-colorings of small outerplanar graphs with $\Delta\leq3$:\\
       a menagerie}
\author{}
\date{}

\newtheorem*{prop}{Proposition}

\begin{document}
\maketitle

\section*{Setup}

For a graph $G$ write $P_e(G,k)$ for the number of proper $k$-edge-colorings of
$G$ (= the chromatic polynomial of the line graph $L(G)$ evaluated at $k$).
Throughout this note $\Delta(G)$ denotes the maximum degree of $G$; we are
interested in $k=3$ and $\Delta(G)\le3$, with $G$ outerplanar.

There is no \emph{universal} closed form for $P_e(G,3)$ on the class of
subcubic outerplanar graphs, but the class is small enough that every $G$ in
it decomposes along its block--cut tree into building blocks each of which
admits a closed-form count.  The building blocks form a short menagerie.

\section*{The menagerie}

\subsection*{1.~Path $P_n$ \;($n$ vertices, $n-1$ edges; $\Delta\le 2$)}

\begin{center}
\includegraphics[width=0.55\textwidth]{fig_path.png}
\end{center}

The line graph $L(P_n)$ is the path $P_{n-1}$, so
\[
P_e(P_n,k) \;=\; P_{\text{vert}}(P_{n-1},k) \;=\; k(k-1)^{n-2},
\qquad
P_e(P_n,3) \;=\; 3\cdot 2^{n-2}.
\]
(For a single edge, $n=2$, the count is $3$; for two edges in a row, $n=3$,
the count is $6$.)

\subsection*{2.~Cycle $C_n$ \;($\Delta=2$)}

\begin{center}
\includegraphics[width=0.32\textwidth]{fig_cycle.png}
\end{center}

The line graph $L(C_n)=C_n$, so
\[
P_e(C_n,k) \;=\; (k-1)^n + (-1)^n(k-1),
\qquad
P_e(C_n,3) \;=\; 2^n + 2(-1)^n.
\]
For even $n$ the count is $2^n+2$; for odd $n$ it is $2^n-2$.

\subsection*{3.~Star $K_{1,3}$ \;(a single $\Delta=3$ vertex)}

\begin{center}
\includegraphics[width=0.25\textwidth]{fig_star.png}
\end{center}

Three pairwise-incident edges at a single vertex must carry three distinct
colors, so $P_e(K_{1,3},3) = 3! = 6$.  More generally
$P_e(K_{1,d},k) = k(k-1)\cdots(k-d+1)$, which is positive iff $k\ge d$, i.e.\
iff $k\ge\Delta(G)$.

\subsection*{4.~Corona $C_n\circ K_1$ \;(cycle with one leaf per cycle vertex; $\Delta=3$)}

\begin{center}
\includegraphics[width=0.42\textwidth]{fig_corona.png}
\end{center}

Each cycle vertex $v$ has degree $3$ in $C_n\circ K_1$: its two cycle edges
must carry distinct colors and its leaf must carry the unique remaining third
color.  So the leaf coloring is \emph{forced} by the cycle coloring, and
\[
P_e(C_n\circ K_1,\,3) \;=\; P_e(C_n,3) \;=\; 2^n + 2(-1)^n.
\]
This is the form of the partial tire dual $D(T)$ in the spoke-only case
(with $L=n+m$).

\subsection*{5.~Trees with $\Delta\le 3$}

A tree $T$ on $n$ vertices has $|E(T)|=n-1$ edges; its line graph $L(T)$ is
a \emph{block graph} (every block is a clique).  Edge-color a tree greedily
by processing edges in BFS order from a leaf: when an edge $\{u,v\}$ is added,
the only colors forbidden are those already used on the edges incident to its
already-colored endpoint.  Hence at any vertex of degree $d$, when the $d$-th
edge is added there are exactly $k-(d-1)$ choices.  For $k=3$ and $\Delta\le3$:
\[
P_e(T,3) \;=\; 3\prod_{e\in E(T)\setminus\{e_0\}} (3-d_e),
\]
where $e_0$ is the first edge processed and $d_e$ is the number of already-
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 $\Delta\le 3$: cycle, possibly with a matching of chords}

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}
\includegraphics[width=0.58\textwidth]{fig_theta133.png}
\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
$p, q \in \{2, 3, 4, 5, 6\}$.

\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
$\{(v_0,v_2),\,(v_3,v_7),\,(v_4,v_6)\}$.  Vertices
$v_0, v_2, v_3, v_4, v_6, v_7$ have degree $3$ (chord endpoints) and
$v_1, v_5$ have degree $2$.

\begin{center}
\includegraphics[width=0.70\textwidth]{fig_3chord_example.png}
\end{center}

We compute $P_e(G,3)$ by propagating constraints.  Let $c_0, c_3, c_4$
denote the colors of the chords $(v_0v_2), (v_3v_7), (v_4v_6)$, and
let $e_i = (v_i, v_{i+1})$ for $i = 0, \dots, 7$ (indices mod $8$) be
the cycle edges.

\emph{Step 1 (fix $c_0$).}  Pick the chord $0{-}2$'s color: 3 choices,
say $c_0 = a$.

\emph{Step 2 (cycle edges at $v_0, v_2$).}  At $v_0$ the three edges
$\{e_0, e_7, \text{chord }02\}$ must use three distinct colors, so
$\{c(e_0), c(e_7)\} = \{b, c\}$ where $\{a,b,c\} = \{1,2,3\}$.  Likewise
$\{c(e_1), c(e_2)\} = \{b, c\}$.  The cycle constraint at $v_1$
(\,$c(e_0)\ne c(e_1)$\,) then forces $c(e_1) = c(e_7)$ and so
$c(e_2) = c(e_0)$.  Two choices for the assignment, say $c(e_0)=b$,
giving $e_0=b, e_1=c, e_2=b, e_7=c$.

\emph{Step 3 (propagate to chord $37$).}  At $v_3$: $\{c(e_2), c(e_3),
c_3\}$ distinct.  We have $c(e_2)=b$, and at $v_7$ similarly
$\{c(e_6), c(e_7), c_3\}$ distinct with $c(e_7)=c$.  Hence
$c_3 \ne b$ and $c_3 \ne c$, forcing $c_3 = a$.  Then
$c(e_3) \ne b, a \Rightarrow c(e_3) = c$, and $c(e_6) \ne c, a
\Rightarrow c(e_6) = b$.

\emph{Step 4 (propagate to chord $46$).}  At $v_4$:
$\{c(e_3), c(e_4), c_4\}$ distinct with $c(e_3) = c$; at $v_6$:
$\{c(e_5), c(e_6), c_4\}$ distinct with $c(e_6) = b$.  Hence
$c_4 \ne c, b \Rightarrow c_4 = a$.  Then $c(e_4) = b$ and
$c(e_5) = c$ (the unique remaining colors).

\emph{Step 5 (verify).}  Cycle edges go $b, c, b, c, b, c, b, c$
around the $8$-cycle (alternating, OK since $8$ is even);
all three chords have color $a$; all degree-$3$ vertices see three
distinct colors.

Total count:
\[
\#(\text{colorings}) \;=\; \underbrace{3}_{c_0\in\{1,2,3\}}
\;\cdot\;\underbrace{2}_{\text{which of }\{b,c\}\text{ is }c(e_0)}
\;=\; 6.
\]
This matches \texttt{Sage}'s direct chromatic-polynomial computation:
$P_e(G,3) = 6$.  Up to permutation of the three colors there is a
\emph{unique} proper $3$-edge-coloring of this $G$.

\subsection*{7.~Block--cut decomposition}

\begin{center}
\includegraphics[width=0.55\textwidth]{fig_blocktree.png}
\end{center}

A general subcubic outerplanar graph is a union of cycle-blocks and edge-blocks
glued at cut vertices.  By case~6 each $2$-connected block is a cycle; the
remaining blocks are single edges (i.e.\ tree edges).  At a cut vertex $v$ of
degree $d_v\in\{2,3\}$, the colors of the $d_v$ edges incident to $v$ must be
pairwise distinct.  Counting $P_e(G,3)$ for the whole graph $G$ amounts to
counting colorings of each block independently and then enforcing the
distinct-color constraint at every cut vertex.  For $k=3$ and $\Delta\le3$
this gives
\[
P_e(G,3) \;=\; \prod_{B \text{ cycle block of }G} P_e(B,3) \;\cdot\;
            \prod_{B \text{ edge block of }G} P_e(B,3)
            \;/\; \prod_{v\text{ cut vertex}} (\text{normalization at } v),
\]
where the normalization corrects the over- or under-counting at the
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 $|V(G)|+|E(G)|$.


\end{document}
