Graph coloring is a well-known and well-studied area of graph theory that has many applications. In this dissertation, we look at two generalizations of graph coloring known as list-coloring and sum-list-coloring. In both of these types of colorings, one seeks to first assign palettes of colors to vertices and then choose a color from the corresponding palette for each vertex so that a proper coloring is obtained.
A celebrated result of Thomassen states that every planar graph can be properly colored from any arbitrarily assigned palettes of five colors. This result is known as 5-list-colorability of planar graphs. Albertson asked whether Thomassen\u27s theorem can be extended by precoloring some vertices which are at a large enough distance apart. Hutchinson asked whether Thomassen\u27s theorem can be extended by allowing certain vertices to have palettes of size less than five assigned to them. In this dissertation, we explore both of these questions and answer them in the affirmative for various classes of graphs.
We also provide a catalog of small configurations with palettes of different prescribed sizes and determine whether or not they can always be colored from palettes of such sizes. These small configurations can be useful in reducing certain planar graphs to obtain more information about their structure.
Additionally, we look at the newer notion of sum-list-coloring where the sum choice number is the parameter of interest. In sum-list-coloring, we seek to minimize the sum of varying sizes of palettes of colors assigned the vertices of a graph. We compute the sum choice number for all graphs on at most five vertices, present some general results about sum-list-coloring, and determine the sum choice number for certain graphs made up of cycles
