An Algebraic Criterion for the Choosability of Graphs.

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 $$v\in V(G)$$ . [ABSTRACT FROM AUTHOR]