The approximate graph colouring problem concerns colouring a $k$-colourable graph with $c$ colours, where $c\geq k$. This problem naturally generalises to promise graph homomorphism and further to promise constraint satisfaction problems. Complexity analysis of all these problems is notoriously difficult. In this paper, we introduce two new techniques to analyse the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism problems.

Corrected a mistakes in the proof of Theorem 1.9 and reformulated Lemma 3.26.

The approximate graph colouring problem concerns colouring a $k$-colourable
graph with $c$ colours, where $c\geq k$. This problem naturally generalises
to promise graph homomorphism and further to promise constraint satisfaction
problems. Complexity analysis of all these problems is notoriously difficult.
In this paper, we introduce two new techniques to analyse the complexity of
promise CSPs: one is based on topology and the other on adjunction. We apply
these techniques, together with the previously introduced algebraic approach,
to obtain new NP-hardness results for a significant class of approximate
graph colouring and promise graph homomorphism problems.