March 2 Math Colloquium

by W. D. Wallis, Southern Illinois University

Totally Magic Labelings and Injections

A labeling of a graph with v vertices and e edges is a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v + e. A labeling is called edge-magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and vertex-magic if the sum of an edge label and the labels of the endpoints of the edge is constant. A totally magic labeling is one that is both edge-magic and vertex-magic. Totally magic labelings appear to be very rare.

A totally magic injection of a graph with v vertices and e edges is a generalization of a totally magic labeling, in which the vertices and edges are mapped one-to-one to any positive integers; these labelings arise in the attempt to find totally magic labelings.

We shall briefly outline the properties and applications of various magic labelings, and explore the existence of totally magic injections.