Packing colorings of subcubic outerplanar graphs

Given a graph G and a nondecreasing sequence S = (s(1),..., s(k)) of positive integers, the mapping c : V (G) -. {1,..., 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,..., k)-packing coloring of a graph G is called the packing chromatic number of G, denoted chi(p)(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,..., 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.