Counting orientations of graphs with no strongly connected tournaments

TL;DR

This paper determines the maximum number of orientations of an n-vertex graph avoiding strongly connected K_k subgraphs, showing it is achieved by Turán graphs for most cases and identifying the unique extremal graphs.

Contribution

It proves that Turán graphs uniquely maximize the number of orientations avoiding strongly connected K_k, extending extremal orientation results to broader graph classes.

Findings

Maximum orientations are given by Turán graphs for k ≥ 4.

Unique extremal graphs are Turán graphs, except for small cases.

Exact counts are provided for specific small graphs like K_4.

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 $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, 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 Tur\'an 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 $K_4$.