Hegde, S.M.Kumudakshi2020-03-312020-03-312015Electronic Notes in Discrete Mathematics, 2015, Vol.48, , pp.151-156https://idr.nitk.ac.in/handle/123456789/11422Bloom 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 C?m(r;m) as a digraph with two directed cycles, one with vertices v1,v2,. . .,vr-1,vr,vr+1,. . .,vm and another directed cycle with vertices v1,v21,. . .,vr-11,vr,vr+11,. . .,vm1 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 Z<inf>n</inf>, where n=2m+1 odd and show that the digraph C?m(r;m) is graceful. 2015 Elsevier B.V.Article