Colorings and List Colorings: Contrasts and Similarities

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A(n ordinary) proper coloring of a (hyper)graph is an assignment of colors to the vertices of this hypergraph in such a way that no edge becomes monochromatic. The chromatic number of a hypergraph is the minimum number of colors used in a proper coloring. Finding proper colorings using only few colors can be useful in many applications. The assignment of colors can model resource allocation; the edges represent conflicts in usage of resources. Areas of application include scheduling, database access, assignment of computer registers, data clustering, computer aided design of printed circuits, positional games, DNA sequencing, etc. The more general model of list coloring allows to restrict the colors available for each vertex. Each vertex is assigned a list of available colors. The color for each vertex must be chosen from its list, but the constraints imposed by edges must still be satisfied by the chosen coloring. List coloring can be used to model problems such as extension of colorings of subgraphs. The list chromatic number of the hypergraph is the minimum k such that whenever all lists have size at least k, it is possible to choose a proper coloring from the assigned lists. A proper coloring with k colors can be considered as a list coloring with all the lists of vertices being the same (of size k). Thus, the list chromatic number of a hypergraph always is at least the chromatic number. Although one might think that the most difficult case of list coloring is when all the lists are the same (giving more conflicts for colors), this is not always the case. Vizing and Erdos, Rubin, and Taylor gave examples of 2 colorable graphs with arbitrarily large list chromatic number. On the other hand, in some classes of (hyper)graphs the list chromatic number behaves similarly to the (ordinary) chromatic number.The main thrust of this project is to explore the analogs for list coloring of many results about ordinary coloring. The issue is under what conditions there are similar bounds for related list coloring and coloring problems. Upper bounds on coloring parameters can lead to efficient algorithms; lower bounds impose limits on what can be accomplished. One question is when the list chromatic number actually equals the chromatic number. Another is how many edges are needed to form a (hyper)graph with a given chromatic number, subject to various additional conditions; here a list coloring analogue will also be explored. For classes of graphs obtained using intersections of geometric objects of special types, bounds on (list) chromatic number in terms of the maximum number of pairwise adjacent vertices are sought. Many coloring problems have analogs using lists, and this project will begin the exploration of many of these list coloring problems. For example, many generalized coloring parameters have bounded values on important classes of graphs such as the planar graphs (those that embed in the plane without edge crossings). Do such colorings still exist when the vertices are assigned color lists of bounded size? Most of the work is planned to be done jointly with Douglas B. West (University of Illinois at Urbana-Champaign)