H-colouring Pt-free graphs in subexponential time

Carla Groenland, Karolina Okrasa, Paweł Rzążewski, Alex Scott, Paul Seymour, Sophie Spirkl

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)184-189
Number of pages6
JournalDiscrete Applied Mathematics
StatePublished - Aug 31 2019

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


  • Colouring
  • P-free
  • Partition function
  • Path-decomposition
  • Subexponential-time algorithm


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