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Minors in graphs of large girth
Author(s) -
Kühn Daniela,
Osthus Deryk
Publication year - 2003
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.10076
Subject(s) - combinatorics , mathematics , conjecture , girth (graph theory) , degree (music) , constant (computer programming) , graph , integer (computer science) , discrete mathematics , physics , computer science , acoustics , programming language
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 C 4 ‐free graphs of sufficiently large chromatic number. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 213–225, 2003