Bounds for the sum choice number

Abstract

A simple graph G=(V,E) is called L-colorable if there is a proper coloring c of the vertices with c(v)∈L(v) for all v∈V where L(v) is an assignment of colors to v. A function f:V→N is called a choice function of G if G is L-colorable for every assignment L with |L(v)|=f(v) for all v∈V. The sum choice number χsc(G) of G is defined as the minimum of ∑v∈Vf(v) over all choice functions f of G.In this note we give general lower and upper bounds for the sum choice number. We determine all connected graphs G whose sum choice number attains the lower bound 2|V|−1 or 2|V|. Moreover, we determine all complete multipartite graphs whose sum choice number attains the upper bound |V|+|E|.