Packing colorings of subcubic graphs

For a sequence S = (s1,…,sk) of non-decreasing positive 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 the distance between u and v is at least si +1, where 1 ≤ i ≤ k. The packing chromatic number, χp(G), of a graph G is defined to be 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 Breˇ sar, Klavˇ zar, Rall, and Wash conjectured that it is true. Furthermore, Gastineau and Togni also asked “Is it true that every subcubic graph except the Petersen graph is packing (1,2,2,2,2,2)-colorable?”. In this talk, we will discuss some recent developments on above-mentioned problems