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The semi-random graph process is an adaptive random graph process in which anonline algorithm is initially presented an empty graph on $n$ vertices. In eachround, a vertex $u$ is presented to the algorithm independently and uniformlyat random. The algorithm then adaptively selects a vertex $v$, and adds theedge $uv$ to the graph. For a given graph property, the objective of thealgorithm is to force the graph to satisfy this property asymptotically almostsurely in as few rounds as possible.  We focus on the property of Hamiltonicity. We present an adaptive strategywhich creates a Hamiltonian cycle in $\alpha n$ rounds, where $\alpha <1.81696$ is derived from the solution to a system of differential equations. Wealso show that achieving Hamiltonicity requires at least $\beta n$ rounds,where $\beta > 1.26575$.

Building Hamiltonian Cycles in the Semi-Random Graph Process in Less Than $2n$ Rounds