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

Abstract

Consider the following problem: Given a transitive tournament $T$ of order $n\ge 3$ and an integer $k$ with $1\le k\le (\genfrac{}{}{0ex}{}{n}{2})$, 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 $k$.