List colorings with distinct list sizes, the case of complete bipartite graphs

A 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 f such that G is f-choosable. In this paper we show that χsc(G)/|V(G)| can be bounded while the minimum degree δmin(G)→∞. (This is not true for the list chromatic number, χℓ(G)) tool is to give tight estimates for the sum choice number for the complete bipartite graphs Ka,q.