Rainbow Hamilton cycles in random graphs and hypergraphs

Let H be an edge colored hypergraph. We say that H contains a rainbow copy of a hypergraph S if it contains an isomorphic copy of S with all edges of distinct colors. We consider the following setting. A randomly edge colored random hypergraph H ∼H k c (n, p) is obtained by adding each k-subset of [n] with probability p, and assigning it a color from [c] uniformly, independently at random. As a first result we show that a typical H ∼ H 2 c (n, p) (that is, a random edge colored graph) contains a rainbow Hamilton cycle, provided that c = (1+ o(1))n and p = logn+log logn+ω(1) n . This is asymptotically best possible with respect to both parameters, and improves a result of Frieze and Loh. Secondly, based on an ingenious coupling idea of McDiarmid, we provide a general tool for tackling problems related to finding “nicely edge colored” structures in random graphs/hypergraphs. We illustrate the generality of this statement by presenting two interesting applications. In one application we show that a typical H ∼H k c (n, p) contains a rainbow copy of a hypergraph S, provided that c = (1+ o(1))|E(S)| and p is (up to a multiplicative constant) a threshold function for the property of containment of a copy of S. In the second application we show that a typical G ∼H 2 c (n, p) contains (1− o(1))np/2 edge disjoint Hamilton cycles, each of which is rainbow, provided that c =ω(n) and p = ω(logn/n). Asaf Ferber Department of Mathematics, Yale University, and Department of Mathematics, MIT, e-mail: [email protected], [email protected]. Michael Krivelevich School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel, Research supported in part by USA-Israel BSF grant 2010115 and by grant 912/12 from the Israel Science Foundation, e-mail: [email protected].