On the Structure of Diregular Digraphs with Defect 1

The Moore bound for a diregular digraph of degree d and diameter k is M d;k = 1 + d + : : : + d k. It is known that digraphs of order M d;k do not exist for d > 1 and k > 1. In this paper we study digraphs of order M d;k ? 1, that is, digraphs with defect 1, denoted by (d; k)-digraphs. If G is a (d; k)-digraph, then for each vertex v of G there exists a vertex w (called the repeat of v) such that there are two walks of lengths k from v to w. In the case of w = v we call v a selfrepeat. To study the existence of (d; k)-digraphs, we may divide the digraphs into two classes according to whether or not they contain a selfrepeat vertex. For d 3 and k 3 we prove that (d; k)-digraphs contain either no selfrepeats or exactly k selfrepeats. Furthermore, we show that every (d; k)-digraph with k selfrepeats must contain a cycle of length k as well as possibly another (d 1 ; k)-digraph as its subdigraph (where d 1 < d). For diameter 2 we give further conditions for the existence of (d; 2)-digraphs with selfrepeats. Additionally, for small degrees we show that (d; 2)-digraphs exist if and only if all their vertices are selfrepeats.