Improved hardness for H-colourings of G-colourable graphs
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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 k k/2-1 colours for every k≥ 4. This improves the result of Bulín, Krokhin, and Opršal [STOC'19], who gave NP-hardness of colouring k-colourable graphs with 2k-1 colours for k≥ 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≥ 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 Opršal [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.
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