TOTALLY ANTIMAGIC TOTAL LABELING OF COMPLETE BIPARTITE GRAPHS MOHAMMED ALI AHMED, J. BASKAR BABUJEE Abstract. For a graph G =(V,E) of order |V (G)| and size |E(G)| a bijection from the union of the vertex set and the edge set of G into the set {1, 2,..., |V (G)| + |E(G)|} is called a total labeling of G. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all edges incident with that vertex. The edge-weight of an edge is the sum of the label of the edge and the labels of the end vertices of that edge. A total labeling is called edge-antimagic (respectively, vertex-antimagic) if all edge-weights (respectively, vertex-weights) are pairwise distinct. If a total labeling is simultaneously edge-antimagic and vertex-antimagic at the same time, then it is called a totally antimagic total labeling. In this paper we prove that complete bipartite graphs admit totally antimagic total labeling. Mathematics Subject Classification (2010): 05C78 Keywords: complete bipartite graphs, totally antimagic total labeling Article history: Received 9 December 2016 Received in revised form 20 March 2017 Accepted 26 March 2017 1. Introduction In this paper we consider finite, simple and undirected graphs. In 1990, Hartsfield and Ringel [6] introduced the notion of an antimagic labeling of graph. A graph with q edges is called antimagic if its edges can be labeled with 1, 2,...,q without repetition, such that the sums of the labels of the edges incident to each vertex are distinct. They conjectured that every tree except P 2 is antimagic and moreover, every connected graph except P 2 is antimagic. This conjecture was proved true, for all graphs having minimum degree Ω (log |V (G)|) by Alon, etc in [1], for more results about antimagic labeling on graphs see [5]. If G is a graph, then V (G) is the vertex set and E(G) is an edge set of G, respectively. A bijection f : V (G) ∪ E(G) →{1, 2,..., |V (G)| + |E(G)|} is called a total labeling of G. A total labeling is called edge-antimagic, if the edge-weights are all distinct. A total labeling is called vertex-antimagic, if the vertex-weights are all distinct. The notion of edge-antimagic total labeling was introduced by Simanjuntak, Bertault and Miller in [8] as a natural extension of magic valuation defined by Kotzing and Rosa in [7]. Simanjuntak, Bertault and Miller [8] proved that C n ,C 2n ,C 2n+1 ,P 2n and P 2n+1 have edge- antimagic total labeling. And the notion of vertex-antimagic total labeling of graphs was introduced by Baˇ ca, etc in [2], were they proved that paths, cycles and other graphs have vertex-antimagic total labeling. If a graph G with p vertices and q edges possessing a labeling that is simultaneously edge-antimagic total labeling and vertex-antimagic total labeling, then this labeling is called a totally antimagic total labeling, and a graph that admits such a labeling is called totally antimagic total graph. The concept of totally antimagic total labeling was introduced by Baˇ ca, etc in [3], were they proved that paths, cycles, stars, double-stars and wheels are totally antimagic total. This concept was introduced as natural extension of ROMANIAN JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2017, VOLUME 7, ISSUE 1, p.21-28