Minimum degree condition of Berge Hamiltonicity in random 3-uniform hypergraphs

Abstract

A graph H has Hamiltonicity if it contains a cycle which covers each vertex of H. In graph theory, Hamiltonicity is a classical and worth studying problem. In 1952, Dirac proved that any n-vertex graph H with minimum degree at least ?n/2? has Hamiltonicity. In 2012, Lee and Sudakov proved that if p ? logn/n , then asympotically almost surely each n-vertex subgraph of random graph G(n,p) with minimum degree at least (1/2 + o(1))np has Hamiltonicity. In this paper, we exend Dirac?s theorem to random 3-uniform hypergraphs. The random 3-uniform hypergraph model H3(n, p) consists of all 3-uniform hypergraphs on n vertices and every possible edge appears with probability p randomly and independently. We prove that if p ? logn/n2, then asympotically almost surely every n-vertex subgraph of H3(n, p) with minimum degree at least (1/4 + o(1))(n 2)p has Berge Hamiltonicity. The value logn/n2 and constant 1/4 both are best possible.