## Out Set-Magic Digraphs

Publisher : Journal of Discrete Mathematical Sciences and Cryptography

Campus : Coimbatore

School : School of Engineering

Department : Mathematics

Verified : Yes

Year : 2013

Abstract : Abstract Motivated by the papers of Peay , Acharya , and SedlaÄek , we introduce out set-magic indexer of digraphs: Let X be a nonempty set, 2 X denote the power-set of X. As in , given a digraph D with p vertices and q arcs, no-self loops, and parallel arcs, is labeled by assigning to each vertex an element from the set 2 X . An arc (x, y) from a vertex x to y is labeled with f âŠ•(x, y) = f (x) âŠ• f (y), where f (x) and f (y) are the values assigned to x and y, and â€œ âŠ• â€ is the symmetric difference of the sets. Such an assignment is called a set-indexer if f and f âŠ• are injective. A set-indexer f of a digraph D is called an out set-magic indexer if, U xâˆ¼e f âŠ•(e) = X, for all xâˆˆ V(D). A digraph admitting an out set-magic indexer is called an out set-magic digraph. In this paper we give some necessary conditions for a digraph to admit an out set-magic indexer, provide the sharp bounds on the size of a digraph admitting an out set-magic indexer. We prove there exist no digraph admitting a unique out set-magic indexer. Also, we give a construction of an out set-magic indexed digraph from a directed path, directed cycle, tournament, inspoken wheel, and directed wind mill, etc., thereby showing their embeddings.