The Complexity of 3-Colouring H-Colourable Graphs

We study the complexity of approximation on satisfiable instances for graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite graph H. In the context of promise constraint satisfaction problems, Brakensiek and Guruswami conjectured that this hardness result extends to promise graph homomorphism as follows: fix any non-bipartite graph H and another graph G with a homomorphism from H to G, it is NP-hard to find a homomorphism to G from a given H-colourable graph. Arguably, the two most important special cases of this conjecture are when H is fixed to be the complete graph on 3 vertices (and G is any graph with a triangle) and when G is the complete graph on 3 vertices (and H is any 3-colourable graph). The former case is equivalent to the notoriously difficult approximate graph colouring problem. In this paper, we confirm the Brakensiek-Guruswami conjecture for the latter case. Our proofs rely on a novel combination of the universal-algebraic approach to promise constraint satisfaction, that was recently developed by Barto, Bulín and the authors, with some ideas from algebraic topology.