%A Chan, Timothy Fong Nam
%A Grzesik, Andrzej
%A Kráľ, Daniel
%A Noel, Jonathan Andrew
%D 2019
%T Cycles of length three and four in tournaments
%K
%X 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 $d\ge 1/16$.
%U http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1222
%J Acta Mathematica Universitatis Comenianae
%0 Journal Article
%P 533-539%V 88
%N 3
%@ 0862-9544
%8 2019-07-30