Groenland, CarlaOkrasa, KarolinaRzążewski, PawełScott, AlexSeymour, PaulSpirkl, Sophie2022-08-122022-08-122019-08-31https://doi.org/10.1016/j.dam.2019.04.010http://hdl.handle.net/10012/18530The final publication is available at Elsevier via https://doi.org/10.1016/j.dam.2019.04.010 © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A graph is called Pt-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 2O (√tn log(n)) for n = |V (G)| in the class of Pt-free graphs G. As a corollary, we show that the number of 3-colourings of a Pt-free graph G can be found in subexponential time. On the other hand, no subexponential time algorithm exists for 4-colourability of Pt-free graphs assuming the Exponential Time Hypothesis. Along the way, we prove that Pt-free graphs have pathwidth that is linear in their maximum degree.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/colouringPt-freesubexponential-time algorithmpartition functionpath-decompositionArticle