Sum choice numbers of some graphs

Let ff be a function assigning list sizes to the vertices of a graph GG. The sum choice number of GG is the minimum ∑v∈V(G)f(v)∑v∈V(G)f(v) such that for every assignment of lists to the vertices of GG, with list sizes given by ff, there exists proper coloring of GG 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 P2□Pn, and the complete bipartite graph K3,nK3,n.