Counting graph orientations with no directed triangles

TL;DR

This paper confirms a conjecture by Alon and Yuster regarding the maximum number of orientations of an n-vertex graph with no directed triangles, establishing the extremal structure as the balanced complete bipartite graph.

Contribution

It proves the conjecture that the maximum is achieved for sufficiently large n and characterizes the unique extremal graphs achieving this maximum.

Findings

Maximum number of orientations is $2^{loor{n^2/4}}$ for large n.

Balanced complete bipartite graph uniquely extremal.

Confirms the conjecture for all n ≥ 8.

Abstract

Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{\lfloor n^2/4\rfloor}$ for $n \geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n \geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{\lfloor n^2/4\rfloor}$ such orientations.