On geometric properties of directed vertex-symmetric graphs

Let Γ be a directed locally finite vertex-symmetric graph such that the underlying graph Γ ¯ is connected. We prove that, in the case Γ is infinite, there exists a positive integer k , depending only on min { deg + ( Γ ) , deg − ( Γ ) } , such that for any positive integer n there exists a directed path of Γ of length not greater than k n 2 whose initial and terminal vertices are at distance n in the graph Γ ¯ . We also prove an analogous result in the case Γ is finite and n < diam ( Γ ¯ ) 2 + 1 . The results are deduced from certain isoperimetric type inequalities for cones in Γ .