Perfect partition of regular bipartite graphs: here is a preprint a honor student wrote on perfect partition a few years ago. There are many open questions to be answered.
If you like linear algebra, you may find this article Orthogonal vector edge coloring interesting. Here is an open question in the article: can you find a graph whose edge-chromatic number is four, while you can assign vectors of three entries to its edges so that incident edges receive orthogonal vectors?
more to come
William & Mary has some funding available for undergraduate research. You may go to Charles Center Research Page to check for such opportunities.
