Directed graphs without short cycles
Combinatorics, Probability and Computing
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A digraph is called k-cyclic if it cannot be made acyclic by removing less than k arcs. It is proved that for every ε 0 there are constants K and δ so that for every d ∈ (0, δn), every ε n2-cyclic digraph with n vertices contains a directed cycle whose length is between d and d + K. A more general result of the same form is obtained for blow-ups of directed cycles.