半随机图过程中 k 选择的力量
Power of $k$ Choices in the Semi-Random Graph Process
AI总结：
本文将半随机图过程推广为每轮从 k 个随机顶点中选择一个进行连边，并研究了在该模型下实现最小度至少为 ℓ、存在完美匹配和存在哈密顿圈这三个单调性质所需的轮数。
摘要：
半随机图过程是一个单人游戏，玩家最初面对一个包含 n 个顶点的空图。在每一轮中，一个顶点 u 独立且均匀随机地呈现给玩家。然后玩家自适应地选择一个顶点 v，并将边 uv 添加到图中。对于一个固定的单调图性质，玩家的目标是以尽可能少的轮数，以高概率迫使图满足该性质。在本文中，我们引入了该游戏的一个自然推广：在每一轮中，向玩家呈现 k 个随机顶点 u_1, …, u_k。她需要选择其中一个被呈现的顶点，并将其连接到任何她想要的顶点。我们关注以下三个单调性质：最小度至少为 ℓ、完美匹配的存在性以及哈密顿圈的存在性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
