Complexity of choosability with a small palette of colors

A graph is $\mathrm{\ell }$-choosable if, for any choice of lists of $\mathrm{\ell }$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs when the number $k$ of colors is limited. We get results which differ surprisingly from the usual case where $k$ is implicit and which extend known results for the usual case. We also exhibit some classes of graphs (defined by structural properties of their blocks) which are choosable.