Reversing Arcs in Transitive Tournaments to Obtain Maximum Number of Spanning Cycles

Abstract

Consider the following problem: Given a transitive tournament \(T\) of order \(n \geq 3\) and an integer \(k\) with \(1 \leq k \leq \binom{n}{2}\), which \(k\) ares in \(T\) should be reversed so that the resulting tournament has the largest number of spanning cycles? In this note, we solve the problem when \(7\) is sufficiently large compared to \(k\).