Extending List Colorings of Planar Graphs

In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix $W$ as a subset of $V(G)$ such that all vertices in $W$ have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where $W=\{u,v\}$ and $G$ has a separating $C_3$ or $C_4$ between $u$ and $v$

Precoloring extension for K4-minor-free graphs

Let G = (V, E) be a graph where every vertex v ∈ V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v) = {1,..., k} for all v ∈ V then a corresponding list coloring is nothing other than an ordinary k-coloring of G. Assume that W ⊆ V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K4-minor-free and d(W) ≥ 7, then such a precoloring of W can be extended to a 4-coloring of all of V. This result clarifies a question posed in [10]. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v) | = 4 for all v ∈ V \ W and d(W) ≥ 7. In both cases the bound for d(W) is best possible

List-coloring and sum-list-coloring problems on graphs

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