Feasible edge colorings of trees with cardinality constraints

A variation of preemptive open shop scheduling corresponds to finding a feasible edge coloring in a bipartite multigraph with some requirements on the size of the different color classes, We show that for trees with fixed maximum degree, one can find in polynomial time an edge k-coloring where for i = 1,,..,k the number of edges of color i is exactly a given number h(i), and each edge e gets its color from a set phi(e) of feasible colors, if such a coloring exists. This problem is NP-complete for general bipartite multigraphs. Applications to open shop problems with costs for using colors are described. (C) 2000 Elsevier Science B.V. All rights reserved.