## Edging Coloring Complete Graphs – Two Examples

Dr. John Caughman from Portland State University presented our Kieval Lectures this year.  He described rainbow graphs and matching, rectangle graphs and lattice paths. I intended to take notes because of past experience but I got so interested that I just went with the flow.  Dr. Caughman is an engaging humorous speaker.  His material was accessible to our students and was well received.  It was very stimulating to think along with his discussions.

One of the concepts was something called rainbow graphs.  I could have spent a little time on the net looking for precise definitions and theorems, but I thought to have fun with my own explorations.

Using Mathematica I tried to create symmetric edge colorings (no edges of the same color meet at the same vertex) of complete graphs that resulted in “nice” matchings.  I used an idea Brian Stonelake suggested of putting a complete graph “inside” or “above” another complete graph of the same size (matching requires an even number of vertices).  I used three dimensions to help me visualize.  The upshot is the two images below and my greater understanding of the intricacy of one aspect of graph theory.

Complete Graph with Six Vertices

Complete Graph with Eight Vertices