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

Abstract

For a sequence $S=(s_1,\dots ,s_k)$ of non-decreasing integers, a packing S-coloring of a graph G is a partition of its vertex set $V\left(G\right)$ into $V_1,\dots ,V_k$ such that for every pair of distinct vertices $u,v\in V_i$, where $1\le i\le k$, the distance between u and v is at least $s_i+1$. The packing chromatic number, ${\chi }_p\left(G\right)$, of a graph G is the smallest integer k such that G has a packing $(1,2,\dots ,k)$-coloring. Gastineau and Togni asked an open question “Is it true that the 1-subdivision ($D\left(G\right)$) 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, ${\chi }_p\left(D\right(G\left)\right)\le 6$ 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.