Constructing integer-magic graphs via the Combinatorial Nullstellensatz

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 construct nontrivial classes of ℤp-magic graphs, prime p ≥ 3. For these graphs, some lower bounds on the number of distinct ℤp-magic labelings are also established.