Sum choice numbers of some graphs

Let f be a function assigning list sizes to the vertices of a graph G. The sum choice number of G is the minimum ∑_v∈V(G)f(v) such that for every assignment of lists to the vertices of G, with list sizes given by f, there exists proper coloring of G from the lists. We answer a few questions raised in a paper of Berliner, Bostelmann, Brualdi, and Deaett. Namely, we determine the sum choice number of the Petersen graph, the cartesian product of paths , and the complete bipartite graph K_3,n.