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Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 67–75 | Cite as

On short cycles in triangle-free oriented graphs

  • Yurong Ji
  • Shufei Wu
  • Hui Song
Article
  • 36 Downloads

Abstract

An orientation of a simple graph is referred to as an oriented graph. Caccetta and Häggkvist conjectured that any digraph on n vertices with minimum outdegree d contains a directed cycle of length at most ⌈n/d⌉. In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that α0 is the smallest real such that every n-vertex digraph with minimum outdegree at least α0n contains a directed triangle. Let ε < (3 − 2α0)/(4 − 2α0) be a positive real. We show that if D is an oriented graph without directed triangles and has minimum outdegree and minimum indegree at least (1/(4 − 2α0)+ε)|D|, then each vertex of D is contained in a directed cycle of length l for each 4 ≤ l < (4 − 2α0)ε|D|/(3 − 2α0) + 2.

Keywords

oriented graph cycle minimum semidegree 

MSC 2010

05C20 05C38 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceHenan Polytechnic UniversityJiaozuo, HenanChina
  2. 2.Center for Discrete MathematicsFuzhou University, Qi Shan Campus of Fuzhou UniversityFujianChina

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