On totally antimagic, edge-magic and vertex-antirnagic total graphs

Abstract

For a graph G = (V,E) of order \V(G)\ and size \E(G)\ a Ejection 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-magic (respectively, edge-antimagic) if all edge-weights are the same (respectively, pairwise distinct). If all the vertex-weights are the same (respectively, pairwise distinct) then the total labeling is called vertex-magic (respectively, vertex-antimagic). If a total labeling is simultaneously edge-antimagic and vertex-antimagic it is called a totally antimagic total labeling. In this paper we prove that complete graphs admit totally antimagic total labeling. We also deal with the problem of finding a total labeling for prism and for two special classes of graphs related to path that are simultaneously edge-magic and vertex-antimagic. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.