Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings

Abstract

Let A be a nontrivial additive 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. Here, the induced label on a vertex is obtained by calculating the sum of the edge labels adjacent to that vertex. 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 Zp-magic labelings (prime p ≥ 3 ) for various graphs, without having to construct the Zp-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 Z3-magic labelings and give some results for various classes of graphs.