Path magic graphs

Let G = (V;E) be a graph with jVj = p and jEj = q. A total labeling l : V [E !f1;2; : : : ; p+qg is a bijection. The weight l : V [E !N is a function induced by l such that l(v) = l(v)+åxv l(vx) and l(e) = l(u)+l(v)+l(e), where e = uv. If l(v) is a constant for all v 2 V, then G is called vertex-magic. If l(e) is a constant for all e 2 E, then G is said to be edge magic. If G is both vertex magic and edge magic, G is called a totally magic graph. In this paper, we define path magic graphs and prove that any star graph is path magic. We also prove that totally magic graphs are path magic, but not the converse.