List colouring is an NP-complete decision problem even if the total number of
colours is three. It is hard even on planar bipartite graphs. We give a
polynomial-time algorithm for solving list colouring of permutation graphs with
a bounded total number of colours. More generally we give a polynomial-time
algorithm that solves the list-homomorphism problem to any fixed target graph
for a large class of input graphs including all permutation and interval
graphs

Enright, Jessica