Graceful digraphs and complete mappings

Abstract

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 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.