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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 $${\{1, 2,\dots,q\}}$${1,2,⋯,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)∪V(H) and E(G+H)=E(G)∪E(H)∪{uv:u∈V(G)andv∈V(H)}. The complete bipartite graph Km,n is an example of join graphs and we give an antimagic labeling for Km,n,n ≥ 2m+1. In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.