List coloring is the following generalization of the graph coloring problem:
Given a graph G = (V, E) and, for every v in V, a list L(v) of colors, is it possible to construct a valid vertex coloring of G such that each vertex v receives a color from the list L(v)?
Note that, in the standard graph coloring problem, all the lists are the same. I will first motivate this problem. Then, I will present two recent results on list coloring. The first is Thomassen's proof that a planar graph can be list-colored if every list has at least 5 colors. The second is Galvin's theorem on list coloring bipartite line graphs.