# Power of $k$ Choices in the Semi-Random Graph Process

## TL;DR

This paper studies a generalized semi-random graph process where multiple vertices are presented each round, analyzing how this affects the ability to quickly achieve properties like high minimum degree, perfect matchings, or Hamiltonian cycles.

## Contribution

It introduces a natural k-choices generalization of the semi-random process and analyzes its impact on achieving key graph properties efficiently.

## Key findings

- Enhanced strategies for k-choice process improve property attainment.
- Quantitative bounds on rounds needed for each property under k choices.
- Comparison with single-choice process shows significant efficiency gains.

## Abstract

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.