Dabrowski, Konrad K. Lozin, Vadim Raman, Rajiv Ries, Bernard 2012 The vertex colouring problem is known to be NP-complete in the class of triangle-free graphs. Moreover, it is NP-complete in any subclass of triangle-free graphs defined by a finite collection of forbidden induced subgraphs, each of which contains a cycle. In this paper, we study the vertex colouring problem in subclasses of triangle-free graphs obtained by forbidding graphs without cycles, i.e., forests, and prove polynomial-time solvability of the problem in many classes of this type. In particular, our paper, combined with some previously known results, provides a complete description of the complexity status of the problem in subclasses of triangle-free graphs obtained by forbidding a forest with at most 6 vertices. application/pdf https://folia.unifr.ch/global/documents/308570 https://folia.unifr.ch/documents/308570/files/triangle-free.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.disc.2011.12.012 info:eu-repo/semantics/openAccess License undefined Discrete Mathematics. - 2012, vol. 312, no. 7, p. 1372-1385 Vertex colouring Triangle-free graphs Polynomial-time algorithm Clique-width info:eu-repo/classification/udc/004 Colouring vertices of triangle-free graphs without forests http://purl.org/coar/resource_type/c_6501