Perfect matchings in random s‐uniform hypergraphs

Abstract

AbstractLet E = {X1, X2…, Xm} where the Xi ⊆ V for 1 ≤ i ≤ m are distinct. The hypergraph G = (V, E) is said to be s‐uniform if |X1| = s for 1 ≤ i ≤ m. A set of edges M = {Xi : i ϵ I} is a perfect matching if (i) i ≠ j ϵ I implies Xi ∩ Xi = 0, and (ii) ∪iϵI Xi = V.In this article we consider the question of whether a random s‐uniform hypergraph contains a perfect matching. Let s ≥ 3 be fixed and m/n4/3 → ∞. We show that an s‐uniform hypergraph with m edges chosen uniformly from [74] contains a perfect matching with high probability. This improves an earlier result of Schmidt and Shamir who showed that m/n3/2 → ∞ suffices. © 1995 John Wiley & Sons, Inc.