On the List Colouring Problem

Abstract

To colour a graph G from lists (L _v)_v∈V(G) is to assign to each vertex v of G one of the colours from its list L _v so that no two adjacent vertices in G are assigned the same colour. The problem, which arises in contexts where all-optical networks are involved, is known to be NP-complete. We are interested in cases where lists of colours are of the same length and show the NP-completeness of the problem when restricted to bipartite graphs (except for lists of length 2, a well known polynomial problem in general). We then show that given any instance of the list colouring problem restricted to lists having the same length l, a solution exists and can be polynomially computed from any k-colouring of the graph, provided that the overall number of available colours does not exceed $k\frac{{\ell - 1}} {{k - 1}}$.