Further results and questions on S-packing coloring of subcubic graphs

Abstract

For a non-decreasing sequence of integers $S=(a_1,a_2,\dots ,a_k)$, an S-packing coloring of G is a partition of $V\left(G\right)$ into k subsets $V_1,V_2,\dots ,V_k$ such that the distance between any two distinct vertices $x,y\in V_i$ is at least $a_i+1$, $1\le i\le k$. We consider the S-packing coloring problem on subclasses of subcubic graphs: For $0\le i\le 3$, 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.