Rainbow Hamilton cycles in random graphs and hypergraphs

Let $H$ be an edge colored hypergraph. We say that $H$ contains a \emph{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 $c=(1+o(1))\mathrm{\mid }E(S)\mathrm{\mid }$ 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 $c=\omega (n)$ and $p=\omega (\log n\mathrm{/}n)$.