  We show how to adjust a very nice coupling argument due to McDiarmid in order
to prove/reprove in a novel way results concerning Hamilton cycles in various
models of random graph and hypergraphs. In particular, we firstly show that for
$k\geq 3$, if $pn^{k-1}/\log n$ tends to infinity, then a random $k$-uniform
hypergraph on $n$ vertices, with edge probability $p$, with high probability
(w.h.p.) contains a loose Hamilton cycle, provided that $(k-1)|n$. This
generalizes results of Frieze, Dudek and Frieze, and reproves a result of
Dudek, Frieze, Loh and Speiss. Secondly, we show that there exists $K>0$ such
for every $p\geq (K\log n)/n$ the following holds: Let $G_{n,p}$ be a random
graph on $n$ vertices with edge probability $p$, and suppose that its edges are
being colored with $n$ colors uniformly at random. Then, w.h.p\ the resulting
graph contains a Hamilton cycle with for which all the colors appear (a rainbow
Hamilton cycle). Lastly, we show that for $p=(1+o(1))(\log n)/n$, if we
randomly color the edge set of a random directed graph $D_{n,p}$ with
$(1+o(1))n$ colors, then w.h.p.\ one can find a rainbow Hamilton cycle where
all the edges are directed in the same way.
