Cordial Labeling of Cartesian Product between two Balanced Bipartite Graphs

Abstract

Cordial Labeling finds its application in Automated Routing algorithms, Communications relevant Adhoc Networks and many others. These networks and routing paths are best represented by a set of vertices and edges which forms a complex graph. In this paper an Algorithmic approach is devised in order to cordially label some of such complex graphs formed by the Cartesian product of two Balanced Bipartite Graphs. There is an absence of a universal algorithm that could label the entire family of \(K_{n,n} \times K_{n,n}\) graphs, which is primarily because of the complexity of these graphs. We have attempted to redefine the \(K_{n,n}\) graph so that on carrying out the Cartesian product we obtain a symmetrical graph. Subsequently we design an efficient and effective algorithm for cordially labeling \(G= K_{n,n} \times K_{n,n}\). We also proof that the algorithm is running with polynomial time complexity.  The results can be used to study and design algorithms for signed product cordial, total signed product cordial, prime cordial labeling etc. of the family of \(K_{n,n} \times K_{n,n}\) graphs.