  We study Hamilton cycles and perfect matchings in a uniform attachment graph.
In this random graph, vertices are added sequentially, and when a vertex $t$ is
created, it makes $k$ independent and uniform choices from $\{1,\dots,t-1\}$
and attaches itself to these vertices. Improving the results of Frieze,
P\'erez-Gim\'enez, Pra\l{}at and Reiniger (2019), we show that, with
probability approaching 1 as $n$ tends to infinity, a uniform attachment graph
on $n$ vertices has a perfect matching for $k \ge 5$ and a Hamilton cycle for
$k\ge 13$. One of the ingredients in our proofs is the identification of a
subset of vertices that is least likely to expand, which provides us with
better expansion rates than the existing ones.
