Counting orientations of graphs with no strongly connected tournaments

Abstract

Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\ge 4$ where $n\ge 5$ or $k\ge 5$, we prove that $S_k(n)=2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4)=40$ and this maximality is achieved only by $K_4$.