Discrete Applied Mathematics (2019) 267 184-189

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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 2Otnlog(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.

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Groenland, C., Okrasa, K., Rzążewski, P., Scott, A., Seymour, P., & Spirkl, S. (2019). H-colouring Pt-free graphs in subexponential time. *Discrete Applied Mathematics*, *267*, 184–189. https://doi.org/10.1016/j.dam.2019.04.010

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