Construction of graceful digraphs using algebraic structures

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, λ) difference set with λ copies of elements of Zv\\ {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 Cm is not graceful for m≡1, 2 (mod 4) we show that the symmetric digraph based on cycle Cm i.e the double cycle, DCm 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).