Colouring Vertices of Triangle-Free Graphs

The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it remains NP-complete even if we additionally exclude a graph F which is not a forest. We study the computational complexity of the problem in (K3, F)-free graphs with F being a forest. From known results it follows that for any forest F on 5 vertices, the vertex colouring problem is polynomial-time solvable in the class of (K3, F)-free graphs. In the present paper, we study the problem for (K3, F)-free graphs with F being a forest on 6 vertices. It is know that in the case when F is the star K1,5, the problem is NP-complete. We show that in nearly all other cases the problem is polynomial-time solvable. The only exception is the class of (K3, 2P3)-free graphs for which the complexity status of the problem remains an open question.

• Type: Journal
• Year: 2010
• Where: WG
• Authors: Konrad Dabrowski, Vadim V. Lozin, Rajiv Raman, Bernard Ries
• Keywords: -free Graphs, Forest, NP-complete, Theoretical Computer Science, WG 2010