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

For a sequence S= (s₁, , sₖ) of non-decreasing integers, a packing S-coloring of a graph G is a partition of its vertex set V (G) into V₁, , Vₖ such that for every pair of distinct vertices u, v Vᵢ, where 1 i k, the distance between u and v is at least sᵢ+1. The packing chromatic number, 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 Bresar, Klavzar, 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, 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.