A generalization of magic and antimagic labelings of graphs

Abstract

A k-magic labeling of a finite, simple graph with |V(G)|=p and |E(G)|=q, is a bijection from the set of edges into the integers {1,2,3,…,q} such that the vertex set V can be partitioned into k sets V1,V2,V3,…,Vk,1≤k≤p, and each vertex in the set Vi has the same vertex sum and any two distinct vertices in different sets have different vertex sum, where a vertex sum is the sum of the labels of all edges incident with that vertex. A graph is called k-magic if it has k-magic labeling, for some k, 1≤k≤p. The study of k-magic labeling is interesting, since all magic graphs are 1-magic and all antimagic graphs are p-magic. In this paper, we discuss the k-magicness of the Complete bipartite graph Kp,p, where 1≤k≤2p.