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

For 1≤s₁≤s₂≤…≤s_k and a graph G, a packing (s₁,s₂,…,s_k)-coloring of G is a partition of V(G) into sets V₁,V₂,…,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 packing chromatic number, χ_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 χ_p(G). Recently, there was a series of papers on packing (s₁,s₂,…,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.