Forcing a sparse minor

4 years 3 months ago
Forcing a sparse minor
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(Kt) = ct √ 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) 3.895 √ 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). 2010 Mathematics Subject Classification. 05C83, 05C35, 05D40.
Bruce A. Reed, David R. Wood
Added 01 Apr 2016
Updated 01 Apr 2016
Type Journal
Year 2016
Where CPC
Authors Bruce A. Reed, David R. Wood
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