Degree-Magic Labelings on the Join and Composition of Complete Tripartite Graphs - Inpoonjai, Phaisatcha (Communications in Mathematics and Applications, 2019)
A graph is called supermagic if there is a labeling of edges, where all edges are differently labeled with consecutive positive integers such that the sum of the labels of all edges, which are incident to each vertex of this graph, is a constant. A graph \(G\) is called degree-magic if all edges can be labeled by integers \(1,2,… ,|E(G)|\) so that the sum of the labels of the edges which are incident to any vertex \(v\) is equal to \((1+|E(G)|)(v)/2\). Degree-magic graphs extend supermagic regular graphs. In this paper, the necessary and sufficient conditions for the existence of degree-magic labelings of graphs obtained by taking the join and composition of complete tripartite graphs are found.