Partial Colorings of Unimodular Hypergraphs

Combinatorial discrepancy theory asks for vertex-colorings of hypergraphs such that all hyperedges contain all colors in (as good as possible) equal quantity. Unimodular hypergraphs are good in this sense: They (and all their induced subhypergraphs as well) can be two-colored in a way that in each hyperedge the number of vertices in one color deviates from that in the other color by at most one. Note that this means that even cardinality hyperedges are perfectly balanced, whereas odd ones have a deviation of exactly one. This observation raises the question whether one can spare these deviations of one by leaving some vertices uncolored. In this work, we give a complete characterization of when this is possible.