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. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as $L(1,1)$-Labelling). A classical complexity result on Colouring is a well-known dichotomy for $H$-free graphs (a graph is $H$-free if it does not contain $H$ as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and 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.

翻译：（正常）着色若满足任意两个颜色类导出子图分别为森林、星森林或孤立顶点与边的无交并，则分别称为无环着色、星着色或单射着色。因此，所有单射着色均为星着色，所有星着色均为无环着色。对应的判定问题分别为无环着色问题、星着色问题与单射着色问题（后者亦称为$L(1,1)$-标号问题）。着色问题的经典复杂度成果是$H$-自由图（若某图不包含$H$作为导出子图则称为$H$-自由图）领域著名的二分性结论。相比之下，尽管多年来已出现大量算法与结构研究成果，针对无环着色、星着色与单射着色计算复杂度的系统性研究尚属空白。本文对此展开系统研究，针对$H$-自由图上的无环着色、星着色与单射着色问题给出了近乎完整的复杂度分类（每个问题各存在一个开放情形）。此外，当颜色数$k$固定（即不作为输入部分）时，我们给出了完整的复杂度分类。研究表明：在$k$固定的情况下，三类问题具有相同的复杂度特性；但当$k$作为输入部分时，该结论不再成立。为获得部分结果，我们证明了涉及图围长及多重图线图类的更强复杂度结论。