Packing chromatic number of cubic graphs

A packingk-coloring of a graph G is a partition of V(G) into sets V₁,…,V_k such that for each 1≤i≤k the distance between any two distinct x,y∈V_i is at least i+1. The packing chromatic number, χ_p(G), of a graph G is the minimum k such that G has a packing k-coloring. Sloper showed that there are 4-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed k and g≥2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k.