Abstract :
We consider a variation of the list colouring problem in which the lists are required to be sets of
consecutive integers, and the colours assigned to adjacent vertices must differ by at least a fixed
integer s. We introduce and investigate a new parameter τ(G) of a graph G, called the consecutive
choosability ratio and defined to be the ratio of the required list size to the separation s in the
limit as s→∞.
We show that the above limit exists and that, for finite graphs G, τ(G) is rational and is a
refinement of the chromatic number χ(G). We provide general bounds on τ(G), and determine
its value for various classes of graphs including bipartite graphs, circuits, wheels and balanced
complete multipartite graphs. Finally, we explore relationships between τ(G) and the circular
chromatic number χc(G).
