Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings

Let A be a nontrivial abelian group and A* = A \ {0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of Ζp-magic labelings (prime p ≥ 3 ) for various graphs, without having to construct the Ζp-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on Ζ3-magic labelings and give some results for various classes of graphs