Improved bounds for covering hypergraphs

TL;DR

This paper improves bounds on hypergraph coverings, extending classical theorems like Graham-Pollak and Katona-Szemerédi to hypergraphs, and explores related parameters such as partition and matching numbers.

Contribution

It provides new bounds and generalizations for hypergraph coverings, including extensions of well-known theorems to r-uniform hypergraphs and analysis of partition and matching numbers.

Findings

Improved bounds for hypergraph covering numbers.

Extended Katona-Szemerédi theorem to r-uniform hypergraphs.

Analyzed relationships between r-partite covering and partition numbers.

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.