Mathematics Qualifying Exam for Julie Vega.

Title: Chromatic Numbers, Hom Complexes, and Topological Obstructions (or a great way to organize your favorite maps)

Abstract: Imagine a pleasant graph, but in color. Now let’s make it a little more exciting by coloring vertices such that no two adjacent vertices have the same color. The smallest number for which you can do this is called the “chromatic number.” In ’78 Lov`asz used the connectivity of the neighborhood complex to find lower bounds for the chromatic number. Later, he generalized this idea to the Hom Complex, Hom(G, H), which encodes graph homomorphisms between G and H and their relationship. In this talk, following an argument by Babson and Kozlov, we will look at Hom complexes and use Stiefel Whitney classes to find lower bounds on the chromatic number of a graph.