A Dirac-type theorem for Berge cycles in random hypergraphs

A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (v(1), e(1), v(2), ..., v(n), e(n)) of distinct vertices v(1), ..., v(n) and distinct hyperedges e(1), ..., e(n) such that {v(1), v(n)} subset of e(n), and {v(i), v(i+1)} subset of e(i) for every i is an element of [n - 1]. We prove the following Dirac-type theorem about Berge cycles in the binomial random r-uniform hypergraph H-(r)(n,p): for every integer r >= 3, every real gamma > 0 and p >= ln(17r) n/n(r-1) asymptotically almost surely, every spanning subgraph H subset of H-(r)(n,p) with minimum vertex degree delta(1)(H) >= (1/2(r-1) +gamma) p((n)(r-1))contains a Hamilton Berge cycle. The minimum degree condition is asymptotically tight and the bound on p is optimal up to some polylogarithmic factor.