Abstract
A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. We give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on H-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k, the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.