Balanced coloring: Equally easy for all numbers of colors?

Abstract

We investigate the problem to color the vertex set of a hypergraph H = (X,ε) with a fixed number of colors in a balanced manner, i.e., in such a way that all hyperedges contain roughly the same number of vertices in each color (discrepancy problem). We show the following result: Suppose that we are able to compute for each induced subhypergraph a coloring in c1 colors having discrepancy at most D. Then there are colorings in arbitrary numbers c2 of colors having discrepancy at most 11/10 c21D. A c2-coloring having discrepancy at most 11/10 c21D +3c1-k|X|can be computed from (c1 -1)(c2 -1) k colorings in c1 colors having discrepancy at most D with respect to a suitable subhypergraph of H. A central step in the proof is to show that a fairly general rounding problem (linear discrepancy problem in c2 colors) can be solved by computing low-discrepancy c1–colorings.