Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k−colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.

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Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form. MSC2010: 17B67.

For a suitable sequence of Weyl group elements {w (l) } l≥0 we give explicit combinatorial descriptions of the crystals B w (l) (kΛ0) for the Demazure modules V w (l) (kΛ0) of the quantum affine Lie algebra Uq( sl(n)) in terms of extended Young diagrams.