Antimagic Labelings of Join Graphs

Abstract

An antimagic labeling of a graph with q edges is a bijection from the set of edges of the graph to the set of positive integers \({\{1, 2,\dots,q\}}\) such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. The join graph G + H of the graphs G and H is the graph with \({V(G + H) = V(G) \cup V(H)}\) and \({E(G + H) = E(G) \cup E(H) \cup \{uv : u \in V(G) {\rm and} v \in V(H)\}}\). The complete bipartite graph Km,n is an example of join graphs and we give an antimagic labeling for \({K_{m,n}, n \geq 2m + 1}\). In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.