Improved bounds for covering hypergraphs

Abstract: The Graham-Pollak theorem states that at least $n-1$ bicliques are required to partition the edge set of the complete graph on $n$ vertices. In this paper, we provide improvements for the generalizations of coverings of graphs and hypergraphs for some specific multiplicities. We study an extension of Katona Szemer\'edi theorem to $r$-uniform hypergraphs. We also discuss the $r$-partite covering number and matching number and how large the $r$-partite partition number would be in terms of $r$-partite covering number for $r$-uniform hypergraphs.