# The complexity of 3-colouring $H$-colourable graphs

## TL;DR

This paper proves a conjecture about the computational hardness of approximating certain graph homomorphism problems, specifically confirming the difficulty of 3-colouring $H$-colourable graphs in a key case.

## Contribution

It establishes the NP-hardness of approximating homomorphisms to 3-cliques for $H$-colourable graphs, advancing understanding of the complexity of promise graph homomorphism problems.

## Key findings

- Confirmed the Brakensiek-Guruswami conjecture for the case where $G$ is a triangle.
- Developed a novel proof combining universal algebra and algebraic topology.
- Enhanced the theoretical framework for understanding the complexity of graph homomorphism problems.

## Abstract

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 Ne\v{s}et\v{r}il, 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, Bul\'in and the authors, with some ideas from algebraic topology.