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 $H \sim \mathcal{H}_{c}^{k}(n,p)$ 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 $H \sim \mathcal{H}_{c}^{2}(n,p)$ (that is, a random edge-colored graph) contains a rainbow Hamilton cycle, provided that $c = (1 + o(1))n$ and $p = \frac{\log n+\log \log n+\omega (1)} {n}$. 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 $H \sim \mathcal{H}_{c}^{k}(n,p)$ contains a rainbow copy of a hypergraph S, provided that $c = (1 + o(1))\vert E(S)\vert$ 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 $G \sim \mathcal{H}_{c}^{2}(n,p)$ contains $(1 - o(1))np/2$ edge-disjoint Hamilton cycles, each of which is rainbow, provided that c = ω(n) and $p =\omega (\log n/n)$.