In this paper, graph coloring problems are studied, where each vertex receives a set of specified size of colors. These sets can be contiguous, in which case the problem models nonpreemtive scheduling, or arbitrary, which models preemtive scheduling. Given a graph with specified for each vertex the number of colors one has to give it, the paper studies the problem to find a multicoloring that minimizes the number of colors (makespan) or the sum of the maximum color numbers over all vertices (sum-of-completion-times measure). The paper studies these problems for planar graphs, and for partial \(k\)-trees (i.e., graphs of bounded treewidth). NP-hardness results are given, also for some special cases. Several properties of independent interest on multicolorings of graphs are derived. These are used to obtain fully polynomial time approximation schemes for the different versions of the problem, both for partial \(k\)-trees, as for planar graphs.