On geometric properties of directed vertex-symmetric graphs

Let ΓΓ be a directed locally finite vertex-symmetric graph such that the underlying graph Γ¯ is connected. We prove that, in the case ΓΓ is infinite, there exists a positive integer kk, depending only on min{deg+(Γ),deg−(Γ)}, such that for any positive integer nn there exists a directed path of ΓΓ of length not greater than kn2kn2 whose initial and terminal vertices are at distance nn in the graph Γ¯. We also prove an analogous result in the case ΓΓ is finite and n