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Colouring (P_r+P_s)-Free Graphs

Authors Tereza Klimosová, Josef Malík, Tomás Masarík, Jana Novotná, Daniël Paulusma, Veronika Slívová

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Tereza Klimosová
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic
Josef Malík
  • Czech Technical University in Prague, Czech Republic
Tomás Masarík
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic
Jana Novotná
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic
Daniël Paulusma
  • Department of Computer Science, Durham University, Durham, UK
Veronika Slívová
  • Computer Science Institute of Charles University, Prague, Czech Republic

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Tereza Klimosová, Josef Malík, Tomás Masarík, Jana Novotná, Daniël Paulusma, and Veronika Slívová. Colouring (P_r+P_s)-Free Graphs. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 5:1-5:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) subseteq {1,...,k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (P_3+P_4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P_3+P_5)-free graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • vertex colouring
  • H-free graph
  • linear forest


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