Further Results and Questions on S-Packing Coloring of Subcubic Graphs

For non-decreasing sequence of integers S=(a₁,a₂, , aₖ), an S-packing coloring of G is a partition of $V(G)$ into k subsets V₁,V₂,,Vₖ such that the distance between any two distinct vertices x,y ∈ Vᵢ is at least aᵢ+1, 1≤ i≤ k. We consider the S-packing coloring problem on subclasses of subcubic graphs: For 0≤ i≤ 3, a subcubic graph G is said to be i-saturated if every vertex of degree 3 is adjacent to at most i vertices of degree 3. Furthermore, a vertex of degree 3 in a subcubic graph is called heavy if all its three neighbors are of degree 3, and G is said to be $(3,i)$-saturated if every heavy vertex is adjacent to at most i heavy vertices. We prove that every 1-saturated subcubic graph is $(1,1,3,3)$-packing colorable and $(1,2,2,2,2)$-packing colorable. We also prove that every $(3,0)$-saturated subcubic graph is $(1,2,2,2,2,2)$-packing colorable.