Palette Sparsification for General Uniform Hypergraphs

2026-08-20

AI Summary

本文证明了对于一般r-均匀超图，通过从大小为αΔ^(1/(r-1))的调色板中独立抽取大小为O(√log n)的颜色列表，可以解决其着色问题。

Abstract

We prove a palette sparsification theorem for general $r$-uniform hypergraphs. For all sufficiently large $n$, every $r\ge 3$, and every $α\ge 7.1$, we show that an $n$-vertex $r$-uniform hypergraph of maximum degree $Δ$ is w.h.p. colorable from independently sampled lists of size $O(\sqrt{\log n})$ drawn from an ambient palette of size $\lceil αΔ^{1/(r-1)}\rceil$. The $\sqrt{\log n}$ dependence is asymptotically tight.

Problem

• palette sparsification
• uniform hypergraphs
• maximum degree
• independently sampled lists

Innovation

• palette sparsification
• uniform hypergraphs
• independent sampling
• asymptotically tight