Lower bounds on the sum choice number of a graph

Given a simple graph G=(V,E)G=(V,E) and a function f from V to the positive integers, f is called a choice function of G if there is a proper vertex coloring ϕ   such that ϕ(v)∈L(v)ϕ(v)∈L(v) for all v∈Vv∈V, where L(v)L(v) is any assignment of f(v)f(v) colors to v. The sum choice number  χsc(G)χsc(G) of G   is defined to be the minimum of ∑v∈Vf(v)∑v∈Vf(v) over all choice functions f of G  . In this paper we provide several new lower bounds on χsc(G)χsc(G) in terms of subgraphs of G.