Improved bounds for randomly colouring simple hypergraphs
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We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ > 0, if k ≥20(1+δ)/δ and q ≥ 100Δ^2+δ/k-4/δ-4, the running time of our algorithm is Õ(poly(Δ k)· n^1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Voung, 2021; He, Sun, and Wu, 2021), and does not require Ω(log n) colours unlike the work of Frieze and Anastos (2017).
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