Dichotomies properties on computational complexity of $S$-packing coloring problems

Abstract

This work establishes the complexity class of several instances of the $S$-packing coloring problem: for a graph $G$, a positive integer $k$ and a nondecreasing list of integers $S=(s_1,\dots ,s_k)$, $G$ is $S$-colorable if its vertices can be partitioned into sets $S_i$, $i=1,\dots ,k$, where each $S_i$ is an $s_i$-packing (a set of vertices at pairwise distance greater than $s_i$). In particular we prove a dichotomy between NP-complete problems and polynomial-time solvable problems for lists of at most four integers.