Institute of Mathematics

On short cycles in triangle-free oriented graphs

Yurong Ji, Shufei Wu, Hui Song

Received March 18, 2016.   First published December 5, 2017.

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 $\lceil n / d\rceil$. In this paper, we consider short cycles in oriented graphs without directed triangles. Suppose that $\alpha_0$ is the smallest real such that every $n$-vertex digraph with minimum outdegree at least $\alpha_0n$ contains a directed triangle. Let $\epsilon< {(3-2\alpha_0)}/{(4-2\alpha_0)}$ 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 $(1/{(4-2\alpha_0)}+\epsilon)|D|$, then each vertex of $D$ is contained in a directed cycle of length $l$ for each $4\le l< {(4-2\alpha_0)\epsilon|D|}/{(3-2\alpha_0)}+2$.
Keywords:  oriented graph; cycle; minimum semidegree
Classification MSC:  05C20, 05C38
DOI:  10.21136/CMJ.2017.0131-16

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Affiliations:   Yurong Ji, School of Mathematics and Information Science, Henan Polytechnic University, Shiji Road, Gaoxin, Jiaozuo 454003, Henan, China, e-mail: jiyurong@hpu.edu.cn; Shufei Wu (corresponding author), School of Mathematics and Information Science, Henan Polytechnic University, Shiji Road, Gaoxin, Jiaozuo 454003, Henan, China; and Center for Discrete Mathematics, Fuzhou University, Qi Shan Campus of Fuzhou University, 2 Xue Yuan Road, University Town, Fuzhou 350003, Fujian, China, e-mail: shufeiwu@hotmail.com; Hui Song, Center for Discrete Mathematics, Fuzhou University, Qi Shan Campus of Fuzhou University, 2 Xue Yuan Road, University Town, Fuzhou 350003, Fujian, China, e-mail: songhuicc@hotmail.com

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