Minors in graphs of large girth

Authors:
Daniela Kühn;Deryk Osthus
Affiliations:
Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany
Venue:
Random Structures & Algorithms
Year:
2003

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Abstract

We show that for every odd integer g ≥ 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr(g+1)/4. This is best possible up to the value of the constant c for g = 5, 7, and 11. More generally, a well-known conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C4-free graphs of sufficiently large chromatic number.