strong $k$-transitive oriented graphs with large minimum degree

A digraph $d=(v,e)$ is $k$-transitive if for any directed $uv$-path of length $k$, we have $(u,v) in e$. in this paper, we study the structure of strong $k$-transitive oriented graphs having large minimum in- or out-degree. we show that such oriented graphs are emph{extended cycles}. as a consequence, we prove that seymour’s second neighborhood conjecture (ssnc) holds for $k$-transitive oriented graphs for $k leq 11$. also we confirm bermond--thomassen conjecture for $k$-transitive oriented graphs for $k leq 11$. a characterization of $k$-transitive oriented graphs having a hamiltonian cycle for $k leq 6$ is obtained immediately.