H-colouring P_t-free graphs in subexponential time

A graph is called P_t-free if it does not contain the path on t vertices as an induced subgraph. Let H be a multigraph with the property that any two distinct vertices share at most one common neighbour. We show that the generating function for (list) graph homomorphisms from G to H can be calculated in subexponential time 2^Otnlog(n) for n=|V(G)| in the class of P_t-free graphs G. As a corollary, we show that the number of 3-colourings of a P_t-free graph G can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of P_t-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that P_t-free graphs have pathwidth that is linear in their maximum degree.