Total Labelings of Graphs with Prescribed Weights

The total labeling of a graph G = (V, E) is a bijection from the union of the vertex set and the edge set of G to the set {1, 2, . . . , |V(G)| + |E(G)|}. The edge-weight of an edge under a total labeling is the sum of the label of the edge and the labels of the end vertices of that edge. The vertexweight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-magic or vertex-magic when all the edge-weights or all the vertex-weights are the same, respectively. When all the edge-weights or all the vertex-weights are different then a total labeling is called edge-antimagic or vertex-antimagic total, respectively. In this paper we deal with the problem of finding a total labeling of some classes of graphs that is simultaneously vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic, respectively. We show several results for stars, paths and cycles.