On the Number of 4-Cycles in a Tournament

Abstract

AbstractIf T is an n-vertex tournament with a given number of 3-cycles, what can be saidabout the number of its 4-cycles? The most interesting range of this problem is whereT is assumed to have c⋅ n 3 cyclic triples for some c> 0 and we seek to minimize thenumber of 4-cycles. We conjecture that the (asymptotic) minimizing T is a randomblow-up of a constant-sized transitive tournament. Using the method of flag algebras,we derive a lower bound that almost matches the conjectured value. We are able toanswer the easier problem of maximizing the number of 4-cycles. These questions canbe equivalently stated in terms of transitive subtournaments. Namely, given the numberof transitive triples in T, how many transitive quadruples can it have? As far as weknow, this is the first study of inducibility in tournaments. 1 Introductionand notation 1.1 Notation For tournaments T,H, let pr(H,T) be the probability that a random set of SHS vertices inT spans a subtournament isomorphic to H. For an infinite family of tournaments T , letpr(H,T ) = lim