Semi-random graph process

TL;DR

This paper introduces a semi-random multigraph process where vertices are randomly selected and connected based on strategic choices, providing bounds on the rounds needed to achieve certain graph properties and approximating various random graph models.

Contribution

It presents a new semi-random graph process, establishes tight bounds for property achievement, and demonstrates the process's ability to approximate existing random graph models.

Findings

Derived tight bounds for property-specific graph formation.

Showed the process can approximate well-known random graph models.

Provided strategies to control the process for desired properties.

Abstract

We introduce and study a novel semi-random multigraph process, described as follows. The process starts with an empty graph on $n$ vertices. In every round of the process, one vertex $v$ of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to $v$. For various natural monotone increasing graph properties $P$, we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies $P$. Along the way, we show that the process is general enough to approximate (using suitable strategies) several well-studied random graph models.