Graceful digraphs and complete mappings

Bloom and Hsu while extending the graceful labelings of graphs to digraphs, specified the relation between graceful unicycles and complete mappings by establishing the relation of each to a particular class of permutations. We denote as a digraph with two directed cycles, one with vertices v1,v2,…,vr−1,vr,vr+1,…,vm and another directed cycle with vertices of same length, such that both the directed cycles have v1 and vr as the two common vertices (where m≥4, 3≤r≤m−1). In this paper we use complete mappings to deduce a partition of Zn, where n=2m+1 odd and show that the digraph is graceful.