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.
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Y. Ji was partially supported by the National Natural Science Foundation of China (No. 11271314). S.Wu was partially supported by the Doctoral Fund of Henan Polytechnic University.
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Ji, Y., Wu, S. & Song, H. On short cycles in triangle-free oriented graphs. Czech Math J 68, 67–75 (2018). https://doi.org/10.21136/CMJ.2017.0131-16
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DOI: https://doi.org/10.21136/CMJ.2017.0131-16