Rainbow Hamilton cycles in random graphs and hypergraphs

Abstract

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 similar to H-c(k) (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 similar to H-c(2)(n,p) (that is, a random edge-colored graph) contains a rainbow Hamilton cycle, provided that c = (1 + o(1))n and p = log n log nlogn co(1) 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 similar to H-c(k)(n,p) contains a rainbow copy of a hypergraph S, provided that c = (1 + o(1))1E(S)1 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 similar to H similar to H-c(2)(n,p) contains (1 o(1))np/2 edge-disjoint Hamilton cycles, each of which is rainbow, provided that c = w(n) and p = omega(log n/n).