The mutual exclusion scheduling problem for permutation and comparability graphs

In this paper, we consider the mutual exclusion scheduling problem for comparability graphs. Given an undirected graph $G$ and a fixed constant $m$, the problem is to find a minimum coloring of $G$ such that each color is used at most $m$ times. The complexity of this problem for comparability graphs was mentioned as an open problem by M\"ohring (1985) and for permutation graphs (a subclass of comparability graphs) as an open problem by Lonc (1991). We prove that this problem is already NP-complete for permutation graphs and for each fixed constant $m \ge 6$