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

Abstract.

 The Moore bound for a diregular digraph of degree E5>, k and diameter k is . It is known that digraphs of order do not exist for d>1 and k>1 ([24] or [6]). In this paper we study digraphs of degree E5>, k, diameter k and order , denoted by (d, k)-digraphs. Miller and Fris showed that (2, k)-digraphs do not exist for k≥3 [22]. Subsequently, we gave a necessary condition of the existence of (3, k)-digraphs, namely, (3, k)-digraphs do not exist if k is odd or if k+1 does not divide [3]. The (E5>, k, 2)-digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d, k)-digraphs. In particular, for , we show that a (d, k)-digraph contains either no cycle of length k or exactly one cycle of length k.