| Balanced k-Colorings (2007) | |||||||||||||||
Abstract | |||||||||||||||
| . While discrepancy theory is normally only studied in the context of 2-colorings, we explore the problem of k-coloring, for k 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 \Gamma 3, where d (the "dimension") is the maximum number of subsets containing a common vertex. For 2colorings, the bound on the discrepancy is at most maxf2d\Gamma3; 2g. Finally, we prove that several restricted versions of computing the discrepancy are NP-complete. 1 Introduction We begin with some basic notation and terminology. Let L be a family of nonempty subsets of a finite set P . We call the elements of P vertices and the elements of L lines. A vertex v 2 P lies on a line ` 2 L if v 2 `. We denote the number of vert... | |||||||||||||||
Publication details | |||||||||||||||
| |||||||||||||||