Construction of graceful digraphs using algebraic structures

Abstract

Abstract: In the early 1980?s Bloom and Hsu extended the notation of graceful labelings to directed graphs, and gave a relationship between graceful digraphs and a variety of algebraic structures. In this paper using a cyclic (v, k, ?) difference set with ? copies of elements of Z<inf>v</inf>\ {0}, we construct graceful digraphs of k vertices and v – 1 arcs. It is known that if gracefully labelled graph has e edges then its symmetric digraph is graceful with the same vertex labels. Although, the cycle C<inf>m</inf> is not graceful for m?1, 2 (mod 4) we show that the symmetric digraph based on cycle C<inf>m</inf> i.e the double cycle, DC<inf>m</inf> which is constructed from a m-cycle by replacing each edge by a pair of arcs, edge xy gives rise to arcs (x, y) and (y, x), is graceful for any m vertices specifically for m?1, 2 (mod 4). © 2016 TARU Publications.