Cycles of length three and four in tournaments

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Timothy Fong Nam Chan Andrzej Grzesik Daniel Kráľ Jonathan Andrew Noel


Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all but one part of equal size and one smaller part. We prove the conjecture for $d\ge 1/36$ by analyzing the possible spectrum of adjacency matrices of tournaments. We also demonstrate that the family of extremal examples is broader than expected and give its full description for $d\ge 1/16$.

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Chan, T., Grzesik, A., Kráľ, D., & Noel, J. (2019). Cycles of length three and four in tournaments. Acta Mathematica Universitatis Comenianae, 88(3), 533-539. Retrieved from