Every subcubic graph is packing (1,1,2,2,3)-colorable

For a sequence S=(s₁,…,s_k) of non-decreasing integers, a packing S-coloring of a graph G is a partition of its vertex set V(G) into V₁,…,V_k such that for every pair of distinct vertices u,v∈V_i, where 1≤i≤k, the distance between u and v is at least s_i+1. The packing chromatic number, χ_p(G), of a graph G is the smallest integer k such that G has a packing (1,2,…,k)-coloring. Gastineau and Togni asked an open question “Is it true that the 1-subdivision (D(G)) of any subcubic graph G has packing chromatic number at most 5?” and later Brešar, Klavžar, Rall, and Wash conjectured that it is true. In this paper, we prove that every subcubic graph has a packing (1,1,2,2,3)-coloring and it is sharp due to the existence of subcubic graphs that are not packing (1,1,2,2)-colorable. As a corollary of our result, χ_p(D(G))≤6 for every subcubic graph G, improving a previous bound (8) due to Balogh, Kostochka, and Liu in 2019, and we are now just one step away from fully solving the conjecture.