Abhishek Dhawan’s scientific contributions

An r-uniform hypergraph $H = (V, E)$ is r-partite if there exists a partition of the vertex set into r parts such that each edge contains exactly one vertex from each part. We say an independent set in such a hypergraph is balanced if it contains an equal number of vertices from each partition. The balanced chromatic number of H is the minimum value q such that H admits a proper q-coloring where each color class is a balanced independent set. In this note, we determine the asymptotic behavior of the balanced chromatic number for sparse r-uniform r-partite Erd\H{o}s--R\'enyi hypergraphs. A key step in our proof is to show that any balanced colorable hypergraph of average degree d admits a proper balanced coloring with $r(r-1)d + 1$ colors. This extends a result of Feige and Kogan on bipartite graphs to this more general setting.