  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.
