Small complete minors above the extremal edge density

Abstract

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 K t -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 K t -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 K t -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.

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Correspondence to Asaf Shapira.

Additional information

Supported in part by NSF Grant DMS-0901355, ISF Grant 224/11 and a Marie-Curie CIG Grant 303320.

Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.

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Shapira, A., Sudakov, B. Small complete minors above the extremal edge density. Combinatorica 35, 75–94 (2015). https://doi.org/10.1007/s00493-015-3013-2

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Mathematics Subject Classication (2000)

  • 05D99
  • 05C83
  • 05C35