In the paper we consider the problems of equitable and semi-equitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semi-equitable coloring is NP- complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unit-length jobs with cubic incompatibility graph on three uniform processors to minimize...

The dissertation discusses the problems of scheduling compatible jobs on parallel machines. Some jobs are incompatible, which is modeled as a binary relation on the set of jobs; the relation is often modeled by an incompatibility graph. We consider two models of machines. The first model, more emphasized in the thesis, is a classical model of scheduling, where each machine does one job at time. The second one is a model of p-batching...

We consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard,...