NP-completeness of graph decomposition problems
Journal of Complexity
Kernels in perfect line-graphs
Journal of Combinatorial Theory Series B
Bounded vertex colorings of graphs
Discrete Mathematics
Restrictions of graph partition problems. Part I
Theoretical Computer Science
Theoretical Computer Science
Selected papers from the second Krakow conference on Graph theory
Restricted coloring models for timetabling
Proceedings of an international symposium on Graphs and combinatorics
Matching and multidimensional matching in chordal and strongly chordal graphs
Discrete Applied Mathematics
Graph classes: a survey
Bounded vertex coloring of trees
Discrete Mathematics
On Complexity of Some Chain and Antichain Partition Problems
WG '91 Proceedings of the 17th International Workshop
The mutual exclusion scheduling problem for permutation and comparability graphs
Information and Computation
Mutual exclusion scheduling with interval graphs or related classes, Part I
Discrete Applied Mathematics
Mutual exclusion scheduling with interval graphs or related classes, Part I
Discrete Applied Mathematics
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This paper is the second part of a study devoted to the mutual exclusion scheduling problem. Given a simple and undirected graph G and an integer k, the problem is to find a minimum coloring of G such that each color is used at most k times. The cardinality of such a coloring is denoted by @g(G,k). When restricted to interval graphs or related classes like circular-arc graphs and tolerance graphs, the problem has some applications in workforce planning. Unfortunately, the problem is shown to be NP-hard for interval graphs, even if k is a constant greater than or equal to four [H.L. Bodlaender, K. Jansen, Restrictions of graph partition problems. Part I. Theoret. Comput. Sci. 148 (1995) 93-109]. In this paper, the problem is approached from a different point of view by studying a non-trivial and practical sufficient condition for optimality. In particular, the following proposition is demonstrated: if an interval graph G admits a coloring such that each color appears at least k times, then @g(G,k)=@?n/k@?. This proposition is extended to several classes of graphs related to interval graphs. Moreover, all our proofs are constructive and provide efficient algorithms to solve the MES problem for these graphs, given a coloring satisfying the condition in input.