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 χ _ρ(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 Δ times ( Δ 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 .