Improved Bounds for Randomly Colouring Simple Hypergraphs

Abstract

We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree ?. For any ? > 0, if k ? 20(1+?)/? and q ? 100?^({2+?}/{k-4/?-4}), the running time of our algorithm is O?(poly(? 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 Vuong, 2021; He, Sun, and Wu, 2021), and does not require ?(log n) colours unlike the work of Frieze and Anastos (2017)