The k L-list τ colouring of a graph G is an L-list colouring (with positive integers) where any two colours assigned to adjacent vertices do not belong to a set τ, where the avoided assignments are listed. Moreover, the length of the list L(x), for every vertex x of G, must be less than or equal to a positive integer k, where k is the number of colours. This problem is NP-complete and we present an efficient heuristic algorithm to solve it. A fundamental aspect of the algorithm we developed is a particular technique of backtracking that permits the direct reassignment of the vertices causing the conflict if, at the moment of assigning a colour to a vertex, no colour on the list associated with it is available. An application of this algorithm to the problem of assigning arriving or leaving trains to the available tracks at a railway station is also discussed.
