Packing Chromatic Number of Subdivisions of Cubic Graphs

Abstract

A packing k-coloring of a graph G is a partition of V(G) into sets $$V_1,\ldots ,V_k$$ such that for each $$1\le i\le k$$ the distance between any two distinct $$x,y\in V_i$$ is at least $$i+1$$ . The packing chromatic number, $$\chi _p(G)$$ , 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 $$\chi _p(G)$$ and of $$\chi _p(D(G))$$ over the class of subcubic graphs G appear in several papers. Gastineau and Togni asked whether $$\chi _p(D(G))\le 5$$ for any subcubic G, and later Bresar, Klavžar, Rall and Wash conjectured this, but no upper bound was proved. Recently the authors proved that $$\chi _p(G)$$ is not bounded in the class of subcubic graphs G. In contrast, in this paper we show that $$\chi _p(D(G))$$ is bounded in this class, and does not exceed 8.