  Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of
positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called
an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in
$c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The
smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring
of a graph $G$ is called the packing chromatic number of $G$, denoted
$\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number
in the class of subcubic (planar) graphs was investigated in several earlier
papers; recently it was established that the invariant is unbounded in the
class of all subcubic graphs.
  In this paper, we prove that the packing chromatic number of any 2-connected
bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove
that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing
coloring, and that there exists a subcubic outerplanar graph with a triangle
that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a
subcubic triangle-free outerplanar graph that does not admit a
$(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite
outerplanar graphs: every such graph admits an $S$-packing coloring for
$S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the
maximum degree of vertices), and this property does not hold if one of the
integers $3$ is replaced by $4$ in the sequence $S$.
