Packing (1,1,2,2)-coloring of some subcubic graphs

For a sequence of non-decreasing positive integers S=(s₁,…,s_k), a packing S-coloring 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 s_i+1. The smallest k such that G has a packing (1,2,…,k)-coloring is called the packing chromatic number of G and is denoted by χ_p(G). For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The question whether χ_p(D(G))≤5 for all subcubic graphs G was first asked by Gastineau and Togni and later conjectured by Brešar, Klavžar, Rall and Wash. Gastineau and Togni observed that if one can prove every subcubic graph except the Petersen graph is packing (1,1,2,2)-colorable then the conjecture holds. The maximum average degree, mad(G), is defined to be max{[Formula presented]:H⊂G}. In this paper, we prove that subcubic graphs with mad(G)<[Formula presented] are packing (1,1,2,2)-colorable. As a corollary, the conjecture of Brešar et al holds for every subcubic graph G with mad(G)<[Formula presented].