Face Anti-magicLabelingsand Deficiency of Graphs

Abstract

Face anti-magic labeling is a significant concept in graph theory that extends the idea of anti-magic labeling from edges and vertices to the faces of planar graphs. A face anti-magic labeling assigns distinct integers to the elements of a graph (vertices, edges, or both) such that the sums associated with each face—typically computed from incident vertices and/or edges—are pairwise distinct. This study explores the existence and construction of face anti-magic labelings for various classes of planar graphs, including cycles, grids, and generalized planar structures.

Furthermore, the notion of deficiency is introduced as a measure of how far a graph is from admitting a face anti-magic labeling. Specifically, the deficiency of a graph is defined as the minimum number of isolated vertices or additional elements required to extend the graph so that it satisfies the face anti-magic condition. We investigate bounds, exact values, and constructive techniques for determining the deficiency of selected graph families.

The results contribute to a deeper understanding of labeling properties in planar graphs and provide new directions for research in combinatorial optimization and graph labeling theory. Potential applications of these findings include network design, frequency assignment, and coding theory, where distinct labeling constraints are essential.