Erdős–Pyber Theorem for Hypergraphs and Secret Sharing

Abstract

A new, constructive proof with a small explicit constant is given to the Erdős–Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or subhypergraphs. In the case every vertex of the graph is connected to at least $n-m$ other vertices, we prove the existence of a fractional covering of the edges by complete bipartite graphs such that every vertex is covered at most $O(m/\log m)$ times, with only a slightly worse explicit constant. This result also generalizes to uniform hypergraphs. Our results give new improved bounds on the complexity of graph and uniform hypergraph based secret sharing schemes, and show the limits of the method at the same time.