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Starting from the trivial observation that, in any optimal coloring of a graph, there always exists a node v such that its neighborhood N(v) contains all colors, we examine related properties in suboptimal colorings (i.e., those using more than (G) colors, where (G) is the chromatic number). In particular, we show that, in any ((G) ؉ p)coloring of G, there is a node v such that its generalized neighborhood N q (v) with q ‫؍‬ max{2p ؊ 1, 2} contains (G) colors for p ≥ 1. Additional properties of ((G) ؉ p)colorings are also given.

We consider the following problem: given suitable integers and p, what is the smallest value such that, for any graph G with chromatic number and any vertex coloring of G with at most + p colors, there is a vertex v such that at least different colors occur within distance of v? Let ( , p) be this value; we show in particular that ( , p) p/2 + 1 for all , p. We give the exact value of when p = 0 or 3, and ( , p) = (4, 1) or (4, 2).

Consider a scheduling problem (P) which consists of a set of jobs to be performed within a limited number of time periods. For each job, we know its duration as an integer number of time periods, and preemptions are allowed. The goal is to assign the required number of time periods to each job while minimizing the assignment and incompatibility costs. When a job is performed within a time period, an assignment cost is encountered, which depends on the involved job and on the considered time period. In addition, for some pairs of jobs, incompatibility costs are encountered if they are performed within common time periods. (P) can be seen as an extension of the multi-coloring problem. We propose various solution methods for (P) (namely a greedy algorithm, a descent method, a tabu search and a genetic local search), as well as an exact approach. All these methods are compared on different types of instances. Keywords Job-scheduling • Multi-coloring • Tabu search • Genetic algorithm 1 Introduction In this paper, we consider a scheduling problem (P) where a set of jobs have to be performed within a limited number of time periods. For each job, we know its duration as an integer number of time periods. Preemptions are allowed (i.e. it is possible to perform a job within non consecutive time periods). Two types of costs are considered: assignment costs and incompatibility costs. When a time period is assigned to a job, an assignment cost is encountered. In addition, for some pairs of jobs, incompatibility costs are encountered if they are performed within common time periods (i.e. the realization of the two jobs overlap in time). The goal is to perform all the jobs at minimum cost. Such a problem can be seen as an extension of the problem studied in (Zufferey et al., 2012), for which all the jobs have a duration of one time period. Problem (P) is a new scheduling problem and there is no literature on it. The reader desiring a review on scheduling models and algorithms is referred to (Pinedo, 2008). Problem (P) can also be seen as an extension of the multicoloring problem (also known as the set-coloring problem, for which there is no cost). Relevant references for the multi-coloring problem with applications in scheduling and in frequency assignment are (Dorne &