NSF East Asia and Pacific Summer Institute (EAPSI) for FY 2013 in Australia

Marshall Emily A, Nashville TN

Investigators

Abstract

This action funds Emily Marshall of Vanderbilt University to conduct a research project in Mathematics during the summer of 2013 at Monash University in Clayton, Victoria, Australia. The project title is "Novel Concepts in Graph Theory." The host scientist is David Wood. The project focuses on several problems in the area of graph minor theory with a particular emphasis on Hamiltonicity. A graph is a set of vertices together with a set of edges which connect some pairs of vertices. A graph H is said to be a minor of a graph G if H can be formed from G by deleting and contracting edges of G. A graph is called Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A graph is called k-connected if the removal of any set of k-1 vertices in the graph never disconnects the graph. The project examines 3-connected graphs with no K2,t minors in regards to Hamiltonicity (where K2,t is the complete bipartite graph with partition sets of size 2 and t), and also looks for a characterization of all K2,5-minor free graphs since a characterization of all K2,4-minor free graphs is already in preparation. In addition, it seeks to determine the minimal forbidden minors of a specific class of minor-closed graphs in which every minor of every graph is subhamiltonian planar. A graph is subhamiltonian planar if it is the spanning subgraph of a planar Hamiltonian graph, and a class of graphs is minor-closed if every minor of every graph in the class is also in the class. A graph is planar if it can be drawn on the plane without crossing edges. Finally, the project examines a conjecture due to Barnette and Goodey involving the Hamiltonicity of 3-connected planar graphs with certain face size restrictions. Broader impacts of an EAPSI fellowship include providing the Fellow a first-hand research experience outside the U.S.; an introduction to the science, science policy, and scientific infrastructure of the respective location; and an orientation to the society, culture and language. These activities meet the NSF goal to educate for international collaborations early in the career of its scientists, engineers, and educators, thus ensuring a globally aware U.S. scientific workforce. Furthermore, the Fellow will have the opportunity to share the results of the project with members of her home institution. The project will strengthen already existing ties between Vanderbilt University and universities in Melbourne paving the way for future international collaboration.