On colouring (2P2,H)-free and (P5,H)-free graphs

Dabrowski, K.K.; Paulusma, D.

Abstract

The Colouring problem asks whether the vertices of a graph can be coloured with at most k colours for a given integer k in such a way that no two adjacent vertices receive the same colour. A graph is (H1,H2)-free if it has no induced subgraph isomorphic to H1 or H2. A connected graph H1 is almost classified if Colouring on (H1,H2)-free graphs is known to be polynomial-time solvable or NP-complete for all but finitely many connected graphs H2. We show that every connected graph H1 apart from the claw K1,3 and the 5-vertex path P5 is almost classified. We also prove a number of new hardness results for Colouring on (2P2,H)-free graphs. This enables us to list all graphs H for which the complexity of Colouring is open on (2P2,H)-free graphs and all graphs H for which the complexity of Colouring is open on (P5,H)-free graphs. In fact we show that these two lists coincide. Moreover, we show that the complexities of Colouring for (2P2,H)-free graphs and for (P5,H)-free graphs are the same for all known cases.

Citation

Dabrowski, K., & Paulusma, D. (2018). On colouring (2P2,H)-free and (P5,H)-free graphs. Information Processing Letters, 134, 35-41. https://doi.org/10.1016/j.ipl.2018.02.003

• Journal Article Type: Article
• Acceptance Date: Feb 2, 2018
• Online Publication Date: Feb 5, 2018
• Publication Date: Jun 1, 2018
• Deposit Date: Feb 5, 2018
• Publicly Available Date: Feb 6, 2018
• Journal: Information Processing Letters
• Print ISSN: 0020-0190
• Electronic ISSN: 1872-6119
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 134
• Pages: 35-41
• DOI: https://doi.org/10.1016/j.ipl.2018.02.003
• Public URL: https://durham-repository.worktribe.com/output/1335526