Currently I am working on crossings numbers of graphs and magic labelings of graphs.

Crossing Numbers of Graphs: One aspect of this research involves the crossing number of the complete graph with n vertices, which is currently unknown for any n>12. One of the standard approaches to this problem involves extending, in all possible ways, all optimal drawings of a complete graph with n-1 or n-2 vertices. I am pursuing a completely different approach to this problem: first proving that an optimal drawing of K_n must have a drawing of K_m that is far from optimal, for m approximately half of n, and then using the limited number of extensions of K_m that could still be in an optimal K_n. This approach has given the first complete proof that cr(K_9)=36, that neither resorts to computer assistance nor any other form of unavailable case-checking. This work, co-authored by R. Bruce Richter is entitled, On the crossing number of K_n without computer assistance, and it will appear in the Journal of Graph Theory. It is available online here. We hope that our recent work will be part of serious new attempts to solve this simple and compelling problem.

Magic Labeling of Graphs: There are many variations of graph labeling problems. Most of my work has focused on Vertex-magic total labelings of graphs. A particularly elegant result, co-authored by Katy Smith, classifies the set of possible magic constants for all odd complete graphs.

One aspect of my research is that I involve undergraduate students into the process as much as possible. This is especially true for my work on vertex-magic total labelings of graphs. Norwich University provides funding for undergraduate research in mathematics, and I have been fortunate enough to participate in many of these projects, some of which have resulted in professional publications with undergraduate students. Here is a nice article about a previous summer research project.

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