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arXiv · 2202.05554

Improved bounds for randomly colouring simple hypergraphs

Weiming Feng·Heng Guo·Jiaheng Wang
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Abstract

We study the problem of sampling almost uniform proper $q$-colourings in $k$-uniform simple hypergraphs with maximum degree $\Delta$. For any $\delta > 0$, if $k \geq\frac{20(1+\delta)}{\delta}$ and $q \geq 100\Delta^{\frac{2+\delta}{k-4/\delta-4}}$, the running time of our algorithm is $\tilde{O}(\mathrm{poly}(\Delta k)\cdot n^{1.01})$, 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 $\Omega(\log n)$ colours unlike the work of Frieze and Anastos (2017).

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Weiming Feng, Heng Guo, Jiaheng Wang. 2022-02-11. Improved bounds for randomly colouring simple hypergraphs. https://arxiv.org/abs/2202.05554

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