A recent palette sparsification theorem of Assadi, Chen, and Khanna [SODA'19] states that in every $n$-vertex graph $G$ with maximum degree $\Delta$, sampling $O(\log{n})$ colors per each vertex independently from $\Delta+1$ colors almost certainly allows for proper coloring of $G$ from the sampled colors. Besides being a combinatorial statement of its own independent interest, this theorem was shown to have various applications to design of algorithms for $(\Delta+1)$ coloring in different models of computation on massive graphs such as streaming or sublinear-time algorithms. In this paper, we further study palette sparsification problems: * We prove that for $(1+\varepsilon) \Delta$ coloring, sampling only $O_{\varepsilon}(\sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper coloring from the sampled colors. * A natural family of graphs with chromatic number much smaller than $(\Delta+1)$ are triangle-free graphs which are $O(\frac{\Delta}{\ln{\Delta}})$ colorable. We prove that sampling $O(\Delta^{\gamma} + \sqrt{\log{n}})$ colors per vertex is sufficient and necessary to obtain a proper $O_{\gamma}(\frac{\Delta}{\ln{\Delta}})$ coloring of triangle-free graphs. * We show that sampling $O_{\varepsilon}(\log{n})$ colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of $(1+\varepsilon) \cdot deg(v)$ arbitrary colors, or even only $deg(v)+1$ colors when the lists are the sets $\{1,\ldots,deg(v)+1\}$. Similar to previous work, our new palette sparsification results naturally lead to a host of new and/or improved algorithms for vertex coloring in different models including streaming and sublinear-time algorithms.

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调色板稀疏化超出 $(\Delta+1)$ 顶点着色

Assadi、Chen 和 Khanna 的最近调色板稀疏定理 [SODA'19] 指出，在每个具有最大度数 $\Delta$ 的 $n$-顶点图 $G$ 中，采样 $O(\log{n})$ 种颜色每个顶点独立于 $\Delta+1$ 颜色几乎可以肯定允许从采样颜色正确着色 $G$。除了作为其自身独立利益的组合陈述之外，该定理还被证明具有多种应用，可用于在大规模图（例如流或次线性时间算法）的不同计算模型中设计 $(\Delta+1)$ 着色算法。在本文中，我们进一步研究了调色板稀疏化问题： * 我们证明对于 $(1+\varepsilon) \Delta$ 着色，每个样本只采样 $O_{\varepsilon}(\sqrt{\log{n}})$ 颜色顶点对于从采样的颜色中获得适当的着色是足够且必要的。* 色数远小于 $(\Delta+1)$ 的自然图族是 $O(\frac{\Delta}{\ln{\Delta}})$ 可着色的无三角形图。我们证明每个顶点采样 $O(\Delta^{\gamma} + \sqrt{\log{n}})$ 颜色对于获得适当的 $O_{\gamma}(\frac{\Delta} {\ln{\Delta}})$ 无三角形图的着色。* 我们表明，每当每个顶点从 $(1+\varepsilon) 的列表中采样时，每个顶点采样 $O_{\varepsilon}(\log{n})$ 颜色足以以高概率正确着色任何图cdot deg(v)$ 任意颜色，或者当列表是集合 $\{1,\ldots,deg(v)+1\}$ 时，甚至只有 $deg(v)+1$ 颜色。与之前的工作类似，