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 P 2 □ P n , and the complete bipartite graph K 3 , n .