Abstract
An orientation of a simple graph is referred to as an oriented graph. Caccetta and Haggkvist 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 e < (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)+e)|D|, then each vertex of D is contained in a directed cycle of length l for each 4 ≤ l < (4 − 2α0)e|D|/(3 − 2α0) + 2.
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