On the packing coloring of undirected and oriented generalized theta graphs

The packing chromatic number χρ(G) of an undirected (respectively, oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V1, . . . , Vk, in such a way that every two distinct vertices in Vi are at distance (respectively directed distance) greater than i in G for every i, 1 ≤ i ≤ k. The generalized theta graph Θ 1,..., p consists of two end-vertices joined by p ≥ 2 internally vertex-disjoint paths with respective lengths 1 ≤ 1 ≤ · · · ≤ p. We prove that the packing chromatic number of any undirected generalized theta graph lies between 3 and max{5, n3+2}, where n3 = |{i / 1 ≤ i ≤ p, i = 3}|, and that both these bounds are tight. We then characterize undirected generalized theta graphs with packing chromatic number k for every k ≥ 3. We also prove that the packing chromatic number of any oriented generalized theta graph lies between 2 and 5 and that both these bounds are tight.