On the Structure of Digraphs with Order Close to the Moore Bound

The Moore bound for a diregular digraph of degree d and diameter k is Md k d d It is known that digraphs of order Md k do not exist for d and k or In this paper we study digraphs of degree d diameter k and order Md k denoted by d k digraphs Miller and Fris showed that k digraphs do not exist for k Subsequently we gave a necessary condition of the existence of k digraphs namely k digraphs do not exist if k is odd or if k does not divide k The d digraphs were considered in In this paper we present further necessary conditions for the existence of d k digraphs In particular for d k we show that a d k digraph contains either no cycle of length k or exactly one cycle of length k