  The semi-random graph process is a single player game in which the player is
initially presented an empty graph on $n$ vertices. In each round, a vertex $u$
is presented to the player independently and uniformly at random. The player
then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For
a fixed monotone graph property, the objective of the player is to force the
graph to satisfy this property with high probability in as few rounds as
possible.
  We focus on the problem of constructing a Hamiltonian cycle in as few rounds
as possible. In particular, we present an adaptive strategy for the player
which achieves it in $\alpha n$ rounds, where $\alpha < 2.01678$ is derived
from the solution to some system of differential equations. We also show that
the player cannot achieve the desired property in less than $\beta n$ rounds,
where $\beta > 1.26575$. These results improve the previously best known bounds
and, as a result, the gap between the upper and lower bounds is decreased from
1.39162 to 0.75102.
