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

翻译：半随机图过程是一种自适应随机图过程，其中在线算法初始面对一个包含 $n$ 个顶点的空图。在每一轮中，一个顶点 $u$ 以独立均匀随机的方式呈现给算法。随后算法自适应地选择一个顶点 $v$，并将边 $uv$ 添加到图中。对于给定的图性质，算法的目标是在尽可能少的轮数内迫使该图以渐近几乎必然的方式满足该性质。我们关注哈密顿性这一性质。我们提出一种自适应策略，能够在 $\alpha n$ 轮内创建一个哈密顿环，其中 $\alpha < 1.81696$ 由微分方程组的解导出。我们还证明了实现哈密顿性至少需要 $\beta n$ 轮，其中 $\beta > 1.26575$。