Packing colorings of subcubic outerplanar graphs

Abstract

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.