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Construction of graceful digraphs using algebraic structures
(Taru Publications, 2016) Hegde, S.M.; Kumudakshi, K.
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 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). © 2016 TARU Publications.
Further Results on Graceful Directed Graphs
(Elsevier B.V., 2016) Hegde, S.M.; Kumudakshi, K.
In this paper we present the gracefulness of the directed graph Pm?Pn? which is an orientation of the planar grid graph Pm?Pn, in which each cell is a unicycle of length four. © 2016 Elsevier B.V.