S-packing colorings of cubic graphs

Given a non-decreasing sequence S = (s(1), s(2),., s(k)) of positive integers, an S-packing coloring of a graph G is a mapping c from V(G) to {si, s2, sk} such that any two vertices with the ith color are at mutual distance greater than s(i), 1 <= i <= k. This paper studies S-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are (1, 2, 2, 2, 2, 2, 2)-packing colorable and (1, 1, 2, 2, 2)-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of a cubic graph of order 38 which is not (1, 2,..., 12)-packing colorable. (C) 2016 Elsevier B.V. All rights reserved.