半随机图过程中的三角饱和图
Triangle-Saturated Graphs in the Semi-Random Graph Process
AI总结：
本文针对半随机图过程的在线与离线版本，研究构造三角饱和图所需轮次的上下界，明确了该图性质下算法的轮次需求范围。
AI中文摘要：
半随机图过程是一种自适应随机图过程：初始时为含n个顶点的空图，每轮向在线算法均匀随机呈现一个顶点u，算法自适应选择顶点v并添加边uv；同时考虑离线版本，即算法在选择前已获得全部随机顶点序列。对于给定图性质，算法目标是用尽可能少的轮次使图渐近几乎必然满足该性质。本文聚焦三角饱和性质，分别为在线和离线版本的过程建立构造三角饱和图所需轮次的上界与下界。
英文摘要：
The semi-random graph process is an adaptive random graph process in which an online algorithm is initially given 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. We also consider the offline version of the process in which the algorithm is given the entire sequence of random vertex choices before the selection takes place. 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. In this paper, we focus on the property of being triangle-saturated and establish upper and lower bounds on the number of rounds required to construct a triangle-saturated graph in both the online and offline versions of the process.
