Abstract
We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is \(\lceil \log _2(n-1)\rceil \). More generally, we also determine the minimum number of orientations of \(K_n\) such that at least one of them orients some specific k-cycles cyclically on every k-element subset of the vertex set. Though only formally related, the questions answered by these results were motivated by an analogous problem of Vera T. Sós concerning triangles and 3-edge-colorings. Some variants of the problem are also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon, N.: Explicit construction of exponential sized families of k-independent sets. Discrete Math. 58, 191–193 (1986)
Alon, N., Spencer, J.H.: The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken (2000)
Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Infinite and Finite Sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), vol. II, pp. 609–627. Colloq. Math. Soc. János Bolyai, vol. 10, North-Holland, Amsterdam (1975)
Fredman, M.L., Komlós, J.: On the size of separating systems and families of perfect hash functions. SIAM J. Algebraic Discrete Methods 5, 61–68 (1984)
Helle, Z.: Anti-Ramsey theorems, MSc. thesis (in Hungarian), Budapest University of Technology and Economics (2007)
Kleitman, D.J., Spencer, J.: Families of \(k\)-independent sets. Discrete Math. 6, 255–262 (1973)
Körner, J., Simonyi, G., Trifference. Studia Sci. Math. Hungar. 30, 95–103 (1995); also in: Deuber, W.A. and Sós, V.T. (eds.) Combinatorics and its Applications to the Regularity and Irregularity of Structures. Akadémiai Kiadó, Budapest (1995)
Körner, J.: Fredman–Komlós bounds and information theory. SIAM J. Algebraic Discrete Methods 7, 560–570 (1986)
Acknowledgments
We are grateful to Benny Sudakov for his inspiring interest. We also thank Gábor Tardos for his useful remarks, and in particular for his help in simplifying the presentation of the proof of Theorem 1. A useful conversation with Kati Friedl is gratefully acknowledged. We also thank an anonymous referee for suggesting the more transparent form upper bound in Proposition 1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by the Hungarian Foundation for Scientific Research (OTKA) Grant Nos. K104343, K105840, and K116769.
Rights and permissions
About this article
Cite this article
Helle, Z., Simonyi, G. Orientations Making k-Cycles Cyclic. Graphs and Combinatorics 32, 2415–2423 (2016). https://doi.org/10.1007/s00373-016-1715-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-016-1715-x