On $S$-packing Coloring of Bounded Degree Graphs

Given a sequence $S=(s_1,s_2,\dots ,s_p)$, $p\ge 2$, of non-decreasing integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $p$ disjoint sets $V_1,\dots ,V_p$ such that any two distinct vertices of $V_i$ are at a distance greater than $s_i$, $1\le i\le p$. In this paper, we study the $S$-packing coloring problem on graphs of bounded maximum degree and for sequences mainly containing 1’s and 2’s ($i^r$ in a sequence means $i$ is repeated $r$ times). Generalizing existing results for subcubic graphs, we prove a series of results on graphs of maximum degree $k$: We show that graphs of maximum degree $k$ are $(1^{k-1},2^k)$-packing colorable. Moreover, we refine this result for restricted subclasses: A graph of maximum degree $k$ is said to be $t$-saturated, $0\le t\le k$, if every vertex of degree $k$ is adjacent to at most $t$ vertices of degree $k$. We prove that any graph of maximum degree $k\ge 3$ is $(1^{k-1},3)$-packing colorable if it is 0-saturated, $(1^{k-1},2)$-packing colorable if it is $t$-saturated, $1\le t\le k-2$; and $(1^{k-1},2^{k-1})$-packing colorable if it is $(k-1)$-saturated. We also propose some conjectures and questions.