Saturation in random graphs

TL;DR

This paper investigates the minimal number of edges in maximal $K_s$-free subgraphs within Erdős-Rényi random graphs, providing asymptotic estimates and revealing unexpected behaviors of these parameters.

Contribution

It offers the first asymptotically tight estimates and exact bounds for minimal edges in maximal $K_s$-free subgraphs of random graphs, extending classical saturation results.

Findings

Asymptotically tight estimates for minimal edges in random graphs

Exact bounds for weak saturation in random graphs

Surprising behaviors of saturation parameters in Erdős-Rényi models

Abstract

A graph $H$ is $K_s$-saturated if it is a maximal $K_s$-free graph, i.e., $H$ contains no clique on $s$ vertices, but the addition of any missing edge creates one. The minimum number of edges in a $K_s$-saturated graph was determined over 50 years ago by Zykov and independently by Erd\H{o}s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal $K_s$-free subgraph of the Erd\H{o}s-R\'enyi random graph $G(n,p)$. We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters.