Every arc-transitive digraph is an orbital digraph

A (simple) digraph is arc-transitive iff it is the orbital digraph for some permutation group. #m/thm/graph

Proof

Let $Γ$ be an arc-transitive digraph and $𝐺 = A u t (Γ)$ . Then for any $(𝛼, 𝛽) \in A (Γ)$ we have $𝐺 (𝛼, 𝛽) = A (Γ)$ . Therefore $Γ$ is an orbital digraph.

Now suppose $𝐺$ acts on $Ω$ and let $Δ = 𝐺 (𝛼, 𝛽)$ be an orbital. Then $(𝛼', 𝛽') \in Δ$ iff $(𝛼', 𝛽') = 𝑔 (𝛼, 𝛽)$ for some $𝑔 \in 𝐺 \leq A u t (Γ)$ , thus $Δ$ is arc-transitive.

#state/tidy | #lang/en | #SemBr