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. In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.

翻译：半随机图过程是一种单人游戏，初始时玩家面对一个包含 $n$ 个顶点的空图。每一轮中，一个顶点 $u$ 独立且均匀随机地呈现给玩家。玩家随后自适应地选择一个顶点 $v$，并将边 $uv$ 添加到图中。对于固定的单调图性质，玩家的目标是在尽可能少的轮数内，以高概率迫使图满足该性质。本文引入了此游戏的一种自然推广：每一轮中，$k$ 个随机顶点 $u_1, \ldots, u_k$ 呈现给玩家，她需要选择其中一个顶点，并连接到任意她想要的顶点。我们聚焦于以下三种单调性质：最小度数至少为 $\ell$、完美匹配的存在性以及哈密顿圈的存在性。