Vertex-primitive digraphs with large fixity

The relative fixity of a digraph \(\Gamma \) $\mathrm{\Gamma }$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of \(\Gamma \) $\mathrm{\Gamma }$ and the number of vertices of \(\Gamma \) $\mathrm{\Gamma }$ . We characterize the vertex-primitive digraphs whose relative fixity is at least \(\frac{1}{3}\) $\frac{1}{3}$ , and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.