Unconditional hardness of approximate graph colouring

Establish unconditional NP-hardness of approximate graph colouring for every pair of constants 3 ≤ k ≤ ℓ, where the input graph is promised to admit a k-colouring and the task is to find an ℓ-colouring.

Background

Approximate graph colouring asks, given a graph that admits a colouring with k colours, for a colouring with ℓ colours, where k ≤ ℓ. The paper explains that the problem is believed to be NP-hard for every constant 3 ≤ k ≤ ℓ, but that known unconditional results cover only restricted parameter ranges. Conditional hardness is known under the d-to-1 conjecture with perfect completeness, whereas proving the full statement unconditionally has remained difficult.

References

It is believed that for every constant $3 \leq k \leq \ell$, this problem remains \NP-hard. While some conditional results are known (i.e.~AGC is \NP-hard if we assume the $d$-to-1 conjecture with perfect completeness), proving unconditional results seems elusive.

— Complexity of approximate conflict-free, linearly-ordered, and nonmonochromatic hypergraph colourings  (2501.12062 - Nakajima et al., 21 Jan 2025) in Section 1, paragraph “Graph colouring”

Deciding 3-colorability is NP-complete, and the promise does not remove the difficulty: coloring a 3-colorable graph with five colors is already NP-hard, and hardness for every constant number of colors is conjectured.

— Coloring 3-colorable graphs with $O(n^{4/23})$ colors via a Gaussian-cover recursion  (2610.01071 - Anand, 1 Oct 2026) in Section 1, Introduction