The mutual exclusion scheduling problem for permutation and comparability graphs

AbstractIn 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 Möhring [Problem 9.10, in: I. Rival (Ed.), Graphs and Orders, Reidel, Dordrecht, 1985, p. 583] and for permutation graphs (a subclass of comparability graphs) as an open problem by Lonc [On complexity of some chain and antichain partition problem, in: G. Schmidt, R. Berghammer (Eds.), Graph Theoretical Concepts in Computer Science, WG 91, Lecture Notes in Computer Science, vol. 570, 1999, pp. 97–104]. We prove that this problem is already NP-complete for permutation graphs and for each fixed constant m⩾6