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 a₈+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.