A game based on vertex-magic total labelings

For a given graph G, let V be the number of vertices and E be the number of edges. Consider a labeling of the vertices and edges which is a one-to-one mapping of the set of integers {1, 2, . . . , V + E} onto the vertices and edges of the graph, with the property that for every vertex the sum of the labels assigned to that vertex and all edges incident with it is some constant k. Although work to date has been on deciding which graphs admit such a labeling (called a vertex-magic total labeling), we shall consider a game based on this concept. For certain families of graphs a winning strategy will be described.