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A Sharp Dirac–Erdős Type Bound for Large Graphs

Published online by Cambridge University Press:  09 March 2018

H. A. KIERSTEAD
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA (e-mail: kierstead@asu.edu)
A. V. KOSTOCHKA
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA Sobolev Institute of Mathematics, Novosibirsk, Russia (e-mail: kostochk@math.uiuc.edu)
A. McCONVEY
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: mcconve2@illinois.edu)

Abstract

Let k ⩾ 3 be an integer, hk(G) be the number of vertices of degree at least 2k in a graph G, and ℓk(G) be the number of vertices of degree at most 2k − 2 in G. Dirac and Erdős proved in 1963 that if hk(G) − ℓk(G) ⩾ k2 + 2k − 4, then G contains k vertex-disjoint cycles. For each k ⩾ 2, they also showed an infinite sequence of graphs Gk(n) with hk(Gk(n)) − ℓk(Gk(n)) = 2k − 1 such that Gk(n) does not have k disjoint cycles. Recently, the authors proved that, for k ⩾ 2, a bound of 3k is sufficient to guarantee the existence of k disjoint cycles, and presented for every k a graph G0(k) with hk(G0(k)) − ℓk(G0(k)) = 3k − 1 and no k disjoint cycles. The goal of this paper is to refine and sharpen this result. We show that the Dirac–Erdős construction is optimal in the sense that for every k ⩾ 2, there are only finitely many graphs G with hk(G) − ℓk(G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and hk(G) − ℓk(G) ⩾ 2k contains k disjoint cycles.

Type
Paper
Copyright
Copyright © Cambridge University Press 2018 

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