Sum list coloring graphs

Let G = (V,E) be a graph with n vertices and e edges. The sum choice number of G is the smallest integer p such that there exist list sizes (f(v):v is an element of V) whose sum is p for which G has a proper coloring no matter which color lists of size f(v) are assigned to the vertices v. The sum choice number is bounded above by n+e. If the sum choice number of G equals n+e, then G is sum choice greedy. Complete graphs K-n are sum choice greedy as are trees. Based on a simple, but powerful, lemma we show that a graph each of whose blocks is sum choice greedy is also sum choice greedy. We also determine the sum choice number of K-2,K-n, and we show that every tree on n vertices can be obtained from K-n by consecutively deleting single edges where all intermediate graphs are sc-greedy.