Improved hardness for H-colourings of G-colourable graphs

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor k/2\rfloor}-1$ colours for every $k\geq 4$. This improves the result of Bul'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness of colouring $k$-colourable graphs with $2k-1$ colours for $k\geq 3$, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring $k$-colourable graphs with $2^{k^{1/3}}$ colours for sufficiently large $k$. Thus, for $k\geq 4$, we improve from known linear/sub-exponential gaps to exponential gaps. Second, we show that the topology of the box complex of H alone determines whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, non-bipartite) G. We use this technique to establish NP-hardness of H-colouring of G-colourable graphs for H that include but go beyond $K_3$, including square-free graphs and circular cliques (leaving $K_4$ and larger cliques open). Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems.

💡 Research Summary

The paper studies approximate graph colourings and, more generally, approximate H‑colourings within the framework of promise constraint satisfaction problems (PCSPs). It presents two major contributions.

First, it dramatically strengthens the hardness of colouring a k‑colourable graph with more than k colours. Previously, the best known NP‑hardness results allowed only a linear or sub‑exponential gap: Bulín, Krokhin and Opršal (STOC’19) proved NP‑hardness of colouring a k‑colourable graph with 2k‑1 colours for any k≥3, while Huang (APPROX‑RANDOM’13) showed NP‑hardness of colouring with 2^{Ω(k^{1/3})} colours for sufficiently large k. The authors improve this to an exponential gap for every k≥4: they prove that colouring a k‑colourable graph with (\binom{k}{\lfloor k/2\rfloor}-1) colours is NP‑hard. This bound is roughly 2^{k·H(1/2)}≈2^{k}, which is exponentially larger than the previous bounds.

The technical core of this result is a simple graph transformation called the arc‑graph (or line‑digraph). This construction reduces the chromatic number in a controlled way, allowing a reduction from the PCSP(K₆, K_{2k}) problem to PCSP(K₄, K_k) for all k≥4. Consequently, the existence of a right‑hard graph (one for which PCSP(G, H) is NP‑hard for every non‑bipartite, loop‑less H) is equivalent to the right‑hardness of K₄. Using Huang’s sub‑exponential hardness as a black‑box, the authors lift it to the exponential regime, obtaining NP‑hardness of PCSP(K_n, K_{⌊n\choose⌊n/2⌋⌋−1}) for all n≥4. For concrete values, the bound improves from 2n−1 (e.g., 11 for n=6) to 19 colours when n=6.

Second, the paper provides a topological characterisation of left‑hardness (hardness for all non‑bipartite G). The key object is the box complex |Box(H)| of a graph H, defined by taking the tensor product H×K₂ and gluing a 2‑dimensional cell to each 4‑cycle. This construction yields a Z₂‑space (a topological space equipped with an involution). Any graph homomorphism G→H induces a Z₂‑equivariant continuous map |Box(G)|→|Box(H)|. The authors prove that left‑hardness depends solely on the existence of Z₂‑maps between box complexes: if H is left‑hard and there is a Z₂‑map from |Box(H₀)| to |Box(H)|, then H₀ is also left‑hard (Theorem 2.7).

Since the box complex of K₃ is homotopy‑equivalent to the circle S¹, any graph whose box complex admits a Z₂‑map to S¹ is left‑hard. This recovers the known result that 3‑colouring of G‑colourable graphs is NP‑hard for all non‑bipartite G, and extends it to broader families such as square‑free graphs (graphs without a 4‑cycle) and circular cliques K_{p/q} with 2<p/q. Thus the paper formalises the intuition that topological obstructions (e.g., Borsuk‑Ulam type arguments) dictate the hardness of H‑colourings.

A unifying methodological contribution is the observation that adjoint functors give reductions between PCSPs. By constructing left and right adjoints between problem instances, the authors obtain reductions that are more flexible than the previously used minion homomorphisms. This perspective underlies the reductions from K₆ to K_{2k} and from K₄ to K_k, and suggests a general algebraic‑topological toolkit for future PCSP hardness proofs.

In summary, the paper achieves (1) an exponential hardness gap for approximate graph colouring, (2) a topological criterion for the hardness of H‑colourings of G‑colourable graphs, and (3) a novel reduction framework based on adjoint functors. These results significantly advance our understanding of the complexity landscape of graph homomorphisms and promise CSPs, and open new avenues for applying algebraic topology to computational hardness.