Balanced k-colorings

While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k greater than or equal to 2, a set of vertices to minimize imbalance among a family of subsets of vertices. The imbalance is the maximum, over all subsets in the family, of the largest difference between the size of any two color classes in that subset. The discrepancy is the minimum possible imbalance. We show that the discrepancy is always at most 4d - 3, where d (the "dimension") is the maximum number of subsets containing a,common vertex. For 2-colorings, the bound on the discrepancy is atmost max{2d - 3,2}. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. (C) 2002 Elsevier Science B.V. All rights reserved.