Packing Chromatic Number of Subdivisions of Cubic Graphs

A packing k-coloring of a graph G is a partition of V(G) into sets $$V_1,\ldots ,V_k$$V1,…,Vk such that for each $$1\le i\le k$$1≤i≤k the distance between any two distinct $$x,y\in V_i$$x,y∈Vi is at least $$i+1$$i+1. The packing chromatic number, $$\chi _p(G)$$χp(G), of a graph G is the minimum k such that G has a packing k-coloring. For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The questions on the value of the maximum of $$\chi _p(G)$$χp(G) and of $$\chi _p(D(G))$$χp(D(G)) over the class of subcubic graphs G appear in several papers. Gastineau and Togni asked whether $$\chi _p(D(G))\le 5$$χp(D(G))≤5 for any subcubic G, and later Brešar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $$\chi _p(G)$$χp(G) is not bounded in the class of subcubic graphs G. In contrast, in this paper we show that $$\chi _p(D(G))$$χp(D(G)) is bounded in this class, and does not exceed 8.