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Forcing a sparse minor

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  16 April 2015

BRUCE REED  and
DAVID R. WOOD
BRUCE REED
Affiliation:
School of Computer Science, McGill University, Montreal, H3A 0E9, Canada and National Institute of Informatics, Japan (e-mail: breed@cs.mcgill.ca)
DAVID R. WOOD
Affiliation:
School of Mathematical Sciences, Monash University, Melbourne, Victoria 3800, Australia (e-mail: david.wood@monash.edu)
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Abstract

This paper addresses the following question for a given graph H: What is the minimum number f(H) such that every graph with average degree at least f(H) contains H as a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth when H is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined f(H) when H has a super-linear number of edges. We focus on the case when H has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if H has t vertices and average degree d at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when H has small average degree, we prove that if H has t vertices and q edges, then f(H) ⩽ t + 6.291q (where the coefficient of 1 in the t term is best possible).

MSC classification

Primary: 05C83: Graph minors
Secondary: 05C35: Extremal problems 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 300 - 322
Copyright
Copyright © Cambridge University Press 2015 

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