Some approaches to a conjecture on short cycles in digraphs

doi:10.1016/S0166-218X(01)00279-7

Discrete Applied Mathematics

Volume 120, Issues 1–3, 15 August 2002, Pages 45-53

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Abstract

We consider the following special case of a conjecture due to Caccetta and Häggkvist: Let D be a digraph on n vertices that all have in-degree and out-degree at least n/3. Then, D contains a directed cycle of length 2 or 3. We discuss several necessary conditions for possible counterexamples to this conjecture, in terms of cycle structure, diameter, maximum degree, clique number, toughness, and local structure. These conditions have not enabled us to prove or refute the conjecture, but they lead to proofs of special instances of the conjecture.

MSC

05C20

05C38

05C35

Keywords

Digraph

Degree condition

Directed triangle

Girth

Cited by (0)

¹: The research was done while the second author was visiting the Faculty of Mathematical Sciences, University of Twente supported by a grant from the Dutch Organization for Scientific Research (NWO) and NSFC.