We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.
@article{arxiv.0912.3802,
title = {The complexity of the list homomorphism problem for graphs},
author = {Laszlo Egri and Andrei Krokhin and Benoit Larose and Pascal Tesson},
journal= {arXiv preprint arXiv:0912.3802},
year = {2010}
}