Cookies

We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Durham University

Research & business

View Profile

Publication details

Dabrowski, K.K. & Paulusma, D. (2018). On colouring (2P2,H)-free and (P5,H)-free graphs. Information Processing Letters 134: 35-41.

Author(s) from Durham

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.