Packing (1,1,2,4)-coloring of subcubic outerplanar graphs

For 1 <= s(1) <= s(2) <= ... <= s(k) and a graph G, a packing (s(1), s(2), ..., s(k))-coloring of G is a partition of V(G) into sets V-1, V-2, ..., 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 packing chromatic number, chi(p)(G), of a graph G is the smallest k such that G has a packing (1, 2, ..., k)-coloring. It is known that there are trees of maximum degree 4 and subcubic graphs G with arbitrarily large chi(p)(G). Recently, there was a series of papers on packing (s(1), s(2), ..., s(k))-colorings of subcubic graphs in various classes. We show that every 2-connected subcubic outerplanar graph has a packing (1, 1, 2)-coloring and every subcubic outerplanar graph is packing (1, 1, 2, 4)-colorable. Our results are sharp in the sense that there are 2-connected subcubic outerplanar graphs that are not packing (1, 1, 3)-colorable and there are subcubic outerplanar graphs that are not packing (1, 1, 2, 5)-colorable. We also show subcubic outerplanar graphs that are not packing (1, 2, 2, 4)-colorable and not packing (1, 1, 3, 4)-colorable. (C) 2021 Elsevier B.V. All rights reserved.