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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Bang-Jensen, G. Z. Gutin: Digraphs: Theory, Algorithms and Applications. Springer Monographs in Mathematics, Springer, London, 2001.
J. A. Bondy: Counting subgraphs a new approach to the Caccetta-Häggkvist conjecture. Discrete Math. 165–166 (1997), 71–80.
L. Caccetta, R. Häggkvist: On minimal digraphs with given girth. Proc. 9th Southeast. Conf. on Combinatorics, Graph Theory, and Computing: Florida Atlantic University. Boca Raton, 1978. Congress. Numer. 21, Utilitas Math. Publishing, Winnipeg, 1978, pp. 181–187.
D. Christofides, P. Keevash, D. Kühn, D. Osthus: A semiexact degree condition for Hamilton cycles in digraphs. SIAM J. Discrete Math. 24 (2010), 709–756.
P. Hamburger, P. Haxell, A. Kostochka: On directed triangles in digraphs. Electron. J. Comb. 14 (2007), Research Paper N19, 9 pages.
J. Hladký, D. Král’, S. Norin: Counting flags in triangle-free digraphs. Extended abstracts of the 5th European Conf. on Combinatorics, Graph Theory and Applications (J. Nešetřil, et al., eds.). Bordeaux, 2009, Elsevier, Amsterdam, Electronic Notes in Discrete Mathematics 34, 2009, pp. 621–625.
P. Keevash, D. Kühn, D. Osthus: An exact minimum degree condition for Hamilton cycles in oriented graphs. J. Lond. Math. Soc., II. Ser. 79 (2009), 144–166.
L. Kelly, D. Kühn, D. Osthus: A Dirac-type result on Hamilton cycles in oriented graphs. Comb. Probab. Comput. 17 (2008), 689–709.
L. Kelly, D. Kühn, D. Osthus: Cycles of given length in oriented graphs. J. Comb. Theory, Ser. B 100 (2010), 251–264.
D. Kühn, D. Osthus: A survey on Hamilton cycles in directed graphs. Eur. J. Comb. 33 (2012), 750–766.
D. Kühn, D. Osthus, A. Treglown: Hamiltonian degree sequences in digraphs. J. Comb. Theory, Ser. B 100 (2010), 367–380.
N. Lichiardopol: A new bound for a particular case of the Caccetta-Häggkvist conjecture. Discrete Math. 310 (2010), 3368–3372.
A. A. Razborov: Flag algebras. J. Symb. Log. 72 (2007), 1239–1282.
J. Shen: Directed triangles in digraphs. J. Comb. Theory, Ser. B 74 (1998), 405–407.
B. D. Sullivan: A summary of results and problems related to the Caccetta-Häggkvist conjecture. Available at ArXiv:math/0605646v1 [math.CO] (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2017.0131-16