Sum List Colorings of Wheels

Abstract

Let $G=(V,E)$ be a simple graph and for every vertex $v\in V$ let $L(v)$ be a set (list) of available colors. The graph $G$ is called $L$ -colorable if there is a proper coloring $\varphi$ of the vertices with $\varphi (v)\in L(v)$ for all $v\in V$. A function $f:V(G) \rightarrow \mathbb N$ is called a choice function of $G$ and $G$ is said to be $f$ -list colorable if $G$ is $L$-colorable for every list assignment $L$ with $|L(v)|=f(v)$ for all $v\in V$. Set ${{\mathrm{size}}}(f)=\sum \nolimits _{v\in V} f(v)$ and define the sum choice number $\chi _{sc}(G)$ as the minimum of ${{\mathrm{size}}}(f)$ over all choice functions $f$ of $G$. It is easy to see that $\chi _{sc}(G)\le |V|+|E|$ for every graph $G$ and that there is a greedy coloring of $G$ for the corresponding choice function $f$ and every list assignment with $|L(v)|=f(v)$. Therefore, a graph $G$ with $\chi _{sc}(G)=|V|+|E|$ is called $sc$ -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 $sc$-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.