Sum List Colorings of Wheels

Abstract

Let $$G=(V,E)$$G=(V,E) be a simple graph and for every vertex $$v\in V$$v?V let $$L(v)$$L(v) be a set (list) of available colors. The graph $$G$$G is called $$L$$L-colorable if there is a proper coloring $$\varphi $$? of the vertices with $$\varphi (v)\in L(v)$$?(v)?L(v) for all $$v\in V$$v?V. A function $$f:V(G) \rightarrow \mathbb N$$f:V(G)?N is called a choice function of $$G$$G and $$G$$G is said to be $$f$$f-list colorable if $$G$$G is $$L$$L-colorable for every list assignment $$L$$L with $$|L(v)|=f(v)$$|L(v)|=f(v) for all $$v\in V$$v?V. Set $${{\mathrm{size}}}(f)=\sum \nolimits _{v\in V} f(v)$$size(f)=?v?Vf(v) and define the sum choice number$$\chi _{sc}(G)$$?sc(G) as the minimum of $${{\mathrm{size}}}(f)$$size(f) over all choice functions $$f$$f of $$G$$G. It is easy to see that $$\chi _{sc}(G)\le |V|+|E|$$?sc(G)≤|V|+|E| for every graph $$G$$G and that there is a greedy coloring of $$G$$G for the corresponding choice function $$f$$f and every list assignment with $$|L(v)|=f(v)$$|L(v)|=f(v). Therefore, a graph $$G$$G with $$\chi _{sc}(G)=|V|+|E|$$?sc(G)=|V|+|E| is called $$sc$$sc-greedy. The concept of the sum choice number was introduced in 2002 by Isaak. In 2006, Heinold characterized the broken wheels (or fan graphs) with respect to $$sc$$sc-greedyness and obtained some results for wheels. In this paper we extend the result for wheels and provide a complete characterization of wheels concerning this property.