Abstract
Abstract
In this paper, we completely resolve the well-known problem of Erdős 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, Rödl and Szemerédi 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 Erdős 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.
Publisher
Cambridge University Press (CUP)
Subject
Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Analysis
Cited by
5 articles.
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