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

For a sequence of non-decreasing positive integers S = (s(1),..., s(k)), a packing S-coloring is a partition of V(G) into sets V-1,...,V-k such that for each 1 <= i <= k the distance between any two distinct x, y is an element of 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 chi(p)(G). For a graph G, let D(G) denote the graph obtained from G by subdividing every edge. The question whether chi(p)(D(G)) <= 5 for all subcubic graphs G was first asked by Gastineau and Togni and later conjectured by Bresar, Klavzar, 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{2 vertical bar E(H)vertical bar/vertical bar V(H)vertical bar : H subset of G}. In this paper, we prove that subcubic graphs with mad(G) < 30/11 are packing (1, 1, 2, 2)-colorable. As a corollary, the conjecture of Bresar et al holds for every subcubic graph G with mad(G) < 30/11. (C) 2020 Elsevier B.V. All rights reserved.