On generalized list colourings of graphs

Vizing [15] and Erdős et al. [8] independently introduce the idea of considering list-colouring and k-choosability. In the both papers the choosability version of Brooks' theorem [4] was proved but the choosability version of Gallai's theorem [9] was proved independently by Thomassen [14] and by Kostochka et al. [11]. In [3] some extensions of these two basic theorems to (,k)-choosability have been proved. In this paper we prove some extensions of the well-known bounds for the -chromatic number to the (,k)-choice number and then an extension of Brooks' theorem.