The vertex colouring problem is known to be NP-comple-te 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 show that the problem is also polynomial-time solvable in many classes of (K3,F)-free graphs with F being a forest on 6 vertices.
