Regular induced subgraphs of a random graph

TL;DR

This paper investigates the size of the largest induced regular subgraph in a typical random graph G(n,1/2), showing it is about n^{2/3} with high probability, addressing a longstanding problem in graph theory.

Contribution

It provides the first probabilistic analysis of the size of the largest induced regular subgraph in random graphs, revealing it scales as n^{2/3}.

Findings

Largest induced regular subgraph in G(n,1/2) is about n^{2/3} in size.

With high probability, no larger induced regular subgraph exists in G(n,1/2).

Addresses a classical problem by analyzing typical case behavior in random graphs.

Abstract

An old problem of Erd\H{o}s, Fajtlowicz and Staton asks for the order of a largest induced regular subgraph that can be found in every graph on n vertices. Motivated by this problem, we consider the order of such a subgraph in a typical graph on n vertices, i.e., in a binomial random graph G(n,1/2). We prove that with high probability a largest induced regular subgraph of G(n,1/2) has about n^{2/3} vertices.