On List Colouring and List Homomorphism of Permutation and Interval   Graphs
  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. 
Jessica Enright, Lorna Stewart, Gabor Tardos
cs.DM