Total labelings of graphs with prescribed weights (Q6586614)

In this paper, the authors discuss edge-magic, vertex-magic, edge-antimagic, and vertex-antimagic total labelings.\N\NThe 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 \(\{ 1, 2, \dots, |V(G)| + |E(G)| \}\). 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 vertex weight 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.\N\NIn this paper, the authors discuss the challenge of finding a complete labeling of some graph classes that are both vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic. They demonstrate the presence of these labels for a few graph types, including cycles, paths, and stars. Two open problems are presented as they wrap up the paper.\N\NThe paper uses the same approach compared with other papers on total labeling for several families of graphs. The examples provided clearly confirm the findings in this paper. The detailed references are appreciated. The paper seems to be a creative piece of work overall. The researchers will benefit a lot from reading the paper.