Uniqueness of Vertex Magic Constants

Dr. Allen O'Neal

General Dynamics C4 Systems

27 January 2012
Shelby Center 218
3:00 (Refreshements at 2:30)

Abstract

In the 2010 version of A Dynamic Survey of Graph Labeling, Gallian references nearly 1200 papers. Gallian states that despite the volume of papers, there are few general results on graph labelings. One type of graph labeling problem is to attempt to label the n vertices of a graph with the integers {1,2,…n} in a fashion such that the sum of the labels in each open neighborhood is a constant. A graph that permits such a labeling is said to be vertex magic, and the constant open neighborhood sum is said to be a vertex magic constant. In this talk we develop the general result that, for all graphs, vertex magic constants are unique and can be determined using the fractional domination number of the graph.