Triangle-Saturated Graphs in the Semi-Random Graph Process

Abstract

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.