Year: 2014
Type: article
Abstract: For every integer t , there is a smallest real number c ( t ) such that any graph with average degree at least c ( t ) must contain a K t -minor (proved by Mader). Improving on results of Shapira and Sudakov, we prove the conjecture of Fiorini, Joret, Theis and Wood that any graph with n vertices and average degree at least c ( t ) + ε must contain a K t -minor consisting of at most C ( ε , t ) log n vertices. Mader also proved that for every integer t , there is a smallest real number s ( t ) such that any graph with average degree larger than s ( t ) must contain a K t -topological minor. We prove that, for sufficiently large t, graphs with average degree at least ( 1 + ε ) s ( t ) contain a K t -topological minor consisting of at most C ( ε , t ) log n vertices. Finally, we show that, for sufficiently large t, graphs with average degree at least ( 1 + ε ) c ( t ) contain either a K t -minor consisting of at most C ( ε , t ) vertices or a K t -topological minor consisting of at most C ( ε , t ) log n vertices.
DOI: 10.1112/jlms/jdu063
Author: Richard Montgomery
Institution: University of Cambridge
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2013272583