Resolution of the Erd\H{o}s-Sauer problem on regular subgraphs

TL;DR

This paper fully resolves the Erd ext{o}s-Sauer problem by establishing that graphs with average degree at least proportional to  log log n contain a k-regular subgraph, improving previous bounds.

Contribution

It proves the exact threshold for the existence of k-regular subgraphs in graphs, matching the lower bound and advancing understanding of regular subgraph existence.

Findings

Graphs with average degree  C_k log log n contain a k-regular subgraph.

Improves previous bound from C_k log n to C_k log log n.

Method applies to problems on almost regular subgraphs and large girth subgraphs.

Abstract

In this paper we completely resolve the well-known problem of Erd\H{o}s and Sauer from 1975 which asks for the maximum number of edges an $n$-vertex graph can have without containing a $k$-regular subgraph, for some fixed integer $k\geq 3$. We prove that any $n$-vertex graph with average degree at least $C_k\log \log n$ contains a $k$-regular subgraph. This matches the lower bound of Pyber, R\"odl and Szemer\'edi and substantially improves an old result of Pyber, who showed that average degree at least $C_k\log n$ is enough. Our method can also be used to settle asymptotically a problem raised by Erd\H{o}s and Simonovits in 1970 on almost regular subgraphs of sparse graphs and to make progress on the well-known question of Thomassen from 1983 on finding subgraphs with large girth and large average degree.