List Colorings with Distinct List Sizes, the Case of Complete Bipartite Graphs

Let f:V→N be a function on the vertex set of the graph G=(V,E). The graph G is f-choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. The sum choice number, χsc(G), is the minimum of ∑f(v), over all functions f such that G is f-choosable. It is known (Alon, Surveys in Combinatorics, 1993 (Keele), London Mathematical Society Lecture Note Series, Vol. 187, Cambridge University Press, Cambridge, 1993, pp. 1-33, Random Struct Algor 16 (2000), 364-368) that if G has average degree d, then the usual choice number χℓ(G) is at least Ω(logd), so they grow simultaneously. In this article, we show that χsc(G)/|V(G)| can be bounded while the minimum degree δmin(G)→∞. Our main tool is to give tight estimates for the sum choice number of the unbalanced complete bipartite graph Ka,q. © 2015 Wiley Periodicals, Inc