Abstract
One of the major and rapidly growing fields of graph theory is hypergraph. A hypergraph is the generalization of the graph. In a simple graph, each edge is identified by 2 end vertices. Also, in a hypergraph, each edge is identified by 1, 2, or more than 2 vertices. Labeling hypergraph plays a major role in many fields. Many researchers define various labeling techniques for hypergraphs. One such labeling is the prime labeling of a hypergraph which defines that each vertex is labeled with a number from 1 to |V| so that the gcd of labels within each hyperedge is 1. The present study examines a new labeling technique namely, relatively prime labeling and relatively prime edge labeling of a hypergraph. Relatively prime labeling states that each vertex is assigned with a distinct number from 1 to |V| so that the labels of vertices of each edge are pairwise relatively prime. Likewise, relatively prime edge labeling of a hypergraph is defined in such a way that each edge is assigned with the labels from 1 to |E|, satisfying the condition that the labels of edges containing each vertex are relatively prime. Also, characterization of relatively prime labeled and relatively prime edge labeled hypergraphs is found in this study. And, for what values of m, n, and k, m-node kuniform hyperpath and m-node k-uniform hypercycle admits relatively prime labeling. Finally, various applications of hypergraph and hypergraph labeling are illustrated with an example. Defining a new labeling technique for hypergraphs with innovative application makes the present study novel for every researcher.
Keywords: Relatively prime labeled hypergraph, Relatively prime edge labeled hypergraph, Hyperpath, Hypercycle, Incidence hypergraph.