Degree conditions forcing oriented cycles

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Roman Glebov Andrzej Grzesik Jan Volec


The longstanding Caccetta-Häggkvist Conjecture is asking for the minimum outdegree (or semidegree) in an oriented graph that forces the appearance of a directed cycle of a bounded length. Motivated by this, Kelly, Kühn and Osthus made a conjecture on the minimal semidegree forcing the appearance of a directed cycle of a given length, and proved it for cycles of length not divisible by 3. Here we prove all the remaining cases of their conjecture with the optimal semidegree threshold.  

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Glebov, R., Grzesik, A., & Volec, J. (2019). Degree conditions forcing oriented cycles. Acta Mathematica Universitatis Comenianae, 88(3), 729-733. Retrieved from