Abstract
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.
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References
Alon, N., Duke, R.A., Lefmann, H., Rödl, V., Yuster, R.: The algorithmic aspects of the Regularity Lemma. J. Algorithms 16, 80–109 (1994)
Alon, N., Shapira, A.: Testing subgraphs in directed graphs. Proc. 35th ACM STOC, ACM Press 700–709 (2003)
Bollobás, B.: Modern Graph Theory. Springer (1998)
Czygrinow, A., Poljak, S., Rödl, V.: Constructive Quasi-Ramsey numbers and tournament ranking. SIAM J. Disc. Math. 12, 48–63 (1999)
de la Vega, W.F.: On the maximal cardinality of a consistent set of arcs in a random tournament. J. Combin. Theory, Ser. B 35, 328f́b-332 (1983)
Komlós, J., Simonovits, M.: Szemerédi regularity lemma and its application in Graph Theory. Paul Erdős is 80, Proc. Coll. Bolyai Math. Soc. Vol 2 (Bolyai Math. Soc., Keszthely) 295–352 (1993)
Szemerédi, E.: Regular partitions of graphs. Proc. Colloque Inter. CNRS 260. (CNRS, Paris) 399–401 (1978)
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Yuster, R. Almost Given Length Cycles in Digraphs. Graphs and Combinatorics 24, 59–65 (2008). https://doi.org/10.1007/s00373-007-0769-1
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DOI: https://doi.org/10.1007/s00373-007-0769-1