An Algebraic Criterion for the Choosability of Graphs

Abstract

Let $G$ be a graph of order $n$ and size $m$. Suppose that $f:V(G)\rightarrow {\mathbb {N}}$ is a function such that $\sum _{v\in V(G)}f(v)=m+n$. In this paper we provide a criterion for $f$-choosability of $G$. Using this criterion, it is shown that the choice number of the complete $k$-partite graph $K_{2,2,\ldots ,2}$ is $k$, which is a well-known result due to Erdös, Rubin and Taylor. Among other results we study the $f$-choosability of the complete $k$-partite graphs with part sizes at most $2$, when $f(v)\in \{k-1,k\}$, for every vertex $v\in V(G)$.