Abstract
A function c:𝑉(𝐺)->{1,2,...,𝑘} is a k-colouring of a graph G if c (u) is not equal to c(v) whenever u and v are adjacent. If any two colour classes induce the disjoint union of vertices and edges, then c is called injective.
Injective colourings are also known as L(1, 1)-labellings and distance 2-colourings. The corresponding decision problem is denoted Injective Colouring.
A graph is H-free if it does not contain H as an induced subgraph. We prove a dichotomy for Injective Colouring for graphs with bounded independence number.
Then, by combining known with further new results, we determine the complexity of Injective Colouring on H-free graphs for every H except for one missing case.