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, χ s c ( G ) , is the minimum of ∑ f ( v ) , over all f such that G is f-choosable. In this paper we show that χ s c ( G ) / | V ( G ) | can be bounded while the minimum degree δ min ( G ) → ∞ . (This is not true for the list chromatic number, χ ℓ ( G ) ) tool is to give tight estimates for the sum choice number for the complete bipartite graphs K a , q .