NOTE - List Colouring when the Chromatic Number is Close to the Order of the Graph

An instance of List Colouring consists of a graph G and a list L(v) of colours for each vertex v of G. We are asked to determine if there is an acceptable colouring of G, that is a colouring in which each vertex receives a colour from its list, and no edge has both its endpoints coloured with the same colour. The list-chromatic number of G, denoted χl(G) is the minimum integer k such that for every assignment of a list L(v) of size at least k to every vertex v of G, there exist an acceptable colouring of G. The list-chromatic number was introduced by Vizing [8], and independently by Erdős et al. [3]. This parameter has received a considerable amount of attention in recent years (see, e.g., [5], [1]). Clearly, by definition, χl(G) ≥ χ(G) because χ(G) = k precisely if an acceptable colouring exists when each L(v) is {1, . . . ,k}. However, the converse inequality is not true, e.g., χ(K3,3) = 3 as can be easily verified by