On the Number of 4-Cycles in a Tournament

Abstract

If T is an n-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where T is assumed to have c ⋅ n 3 cyclic triples for some c > 0 and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing T is a random blow-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 to answer the easier problem of maximizing the number of 4-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in T , how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.

Cycles of length three and four in tournaments

“…Although 0 ≤ t(C 3 , T ) ≤ 1/8 for every tournament T , the conjecture is currently only known to hold for tournaments with 3-cycle density asymptotically equal to 0, 1/8, or 1/32 [9]. Figure 2: The conjectured region of asymptotically feasible densities of C 3 and C 4 in tournaments.…”

“…In this paper, we investigate a related problem for tournaments posed by Linial and Morgenstern [9], who asked for the minimum density of 4-cycles in a large tournament with fixed density of 3-cycles. They conjectured that the tournament asymptotically minimizing this density is a blow-up of a transitive tournament with all but one part of equal size and one smaller part in which the arcs within each part are oriented randomly (they call this construction a random blow-up), i.e., the structure of the conjectured extremal examples is akin to those of the Erdős-Rademacher problem.…”

“…Next, we describe the family of conjectured tight examples from [9] which will motivate the definition of the function g below. Given z ∈ [0, 1], we let n be an integer chosen large with respect to z and define an n-vertex tournament T as follows.…”

“…because of the concentration around the expected values. The conjecture of Linial and Morgenstern [9] asserts that the above construction is asymptotically optimal. In light of this, we write the regime of k parts to denote the set of values of t(C 3 , T ) between 1/(8k 2 ) and 1/(8(k − 1) 2 ), corresponding to the range of values for which the above construction has its vertices split into k parts.…”

A Note on the Inducibility of $$4$$ 4 -Vertex Graphs

“…Indeed, there is substantial recent activity in this domain [20,19,22], and additional combinatorial structures with natural notions of local profile and inducibility are being investigated as well, e.g. tournaments [23] trees [4], and permutations [35].…”