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

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Abstract

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ISAAC.2018.5

Author Details

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

Funding

T. Masařík, J. Novotná and V. Slívová were supported by the project GAUK 1277018 and the grant SVV–2017–260452. T. Klimošová was supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR, by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004), and by the project GAUK 1277018. T. Masařík was also partly supported by the Center of Excellence – ITI, project P202/12/G061 of GA ČR. V. Slívová was partly supported by the project 17-09142S of GA ČR and Charles University project PRIMUS/17/SCI/9. D. Paulusma was supported by the Leverhulme Trust (RPG-2016-258).

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