Small complete minors above the extremal edge density

Asaf Shapira, Benny Sudakov

Research output: Contribution to journalArticlepeer-review


A fundamental result of Mader from 1972 asserts that a graph of high average degree contains a highly connected subgraph with roughly the same average degree. We prove a lemma showing that one can strengthen Mader’s result by replacing the notion of high connectivity by the notion of vertex expansion.

Another well known result in graph theory states that for every integer t there is a smallest real c(t), such that every n-vertex graph with c(t)n edges contains a Kt-minor. Fiorini, Joret, Theis and Wood asked whether every n-vertex graph G that has at least (c(t)+ε)n edges, must contain a Kt-minor of order at most C(ε) logn. We use our extension of Mader’s theorem to prove that such a graph G must contain a Kt-minor of order at most C(ε) lognlognlogn. Known constructions of graphs with high girth show that this result is tight up to the log logn factor.

Original languageEnglish
Pages (from-to)75-94
Number of pages20
Issue number1
StatePublished - 1 Feb 2015


  • 05C35
  • 05C83
  • 05D99

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


Dive into the research topics of 'Small complete minors above the extremal edge density'. Together they form a unique fingerprint.

Cite this