H-colouring Pt-free graphs in subexponential time
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Date
2019-08-31
Authors
Groenland, Carla
Okrasa, Karolina
Rzążewski, Paweł
Scott, Alex
Seymour, Paul
Spirkl, Sophie
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
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.
Description
The 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/
Keywords
colouring, Pt-free, subexponential-time algorithm, partition function, path-decomposition