A seminal palette sparsification result of Assadi, Chen, and Khanna states that in every $n$-vertex graph of maximum degree $\Delta$, sampling $\Theta(\log n)$ colors per vertex from $\{1, \ldots, \Delta+1\}$ almost certainly allows for a proper coloring from the sampled colors. Alon and Assadi extended this work proving a similar result for $O\left(\Delta/\log \Delta\right)$-coloring triangle-free graphs. Apart from being interesting results from a combinatorial standpoint, their results have various applications to the design of graph coloring algorithms in different models of computation. In this work, we focus on locally sparse graphs, i.e., graphs with sparse neighborhoods. We say a graph $G = (V, E)$ is $k$-locally-sparse if for each vertex $v \in V$, the subgraph $G[N(v)]$ contains at most $k$ edges. A celebrated result of Alon, Krivelevich, and Sudakov shows that such graphs are $O(\Delta/\log (\Delta/\sqrt{k}))$-colorable. For any $\alpha \in (0, 1)$ and $k \ll \Delta^{2\alpha}$, let $G$ be a $k$-locally-sparse graph. For $q = \Theta\left(\Delta/\log \left(\Delta^\alpha/\sqrt{k}\right)\right)$, we show that sampling $O\left(\Delta^\alpha + \sqrt{\log n}\right)$ colors per vertex is sufficient to obtain a proper $q$-coloring of $G$ from the sampled colors. Setting $k = 1$ recovers the aforementioned result of Alon and Assadi for triangle-free graphs. A key element in our proof is a proposition regarding correspondence coloring in the so-called color-degree setting, which improves upon recent work of Anderson, Kuchukova, and the author and is of independent interest.

翻译：Assadi、Chen和Khanna的一项开创性调色板稀疏化结果表明：在每个最大度为Δ的n顶点图中，从{1, …, Δ+1}中为每个顶点采样Θ(log n)种颜色，几乎必然可以从采样颜色中获得正常着色。Alon与Assadi扩展了这项工作，证明了无三角形图O(Δ/log Δ)-着色的类似结果。除了组合学意义上的有趣结论外，这些结果在不同计算模型的图着色算法设计中具有多种应用。本文聚焦于局部稀疏图，即具有稀疏邻域的图。若对于每个顶点v∈V，子图G[N(v)]至多包含k条边，则称图G=(V,E)是k-局部稀疏的。Alon、Krivelevich和Sudakov的著名结果表明此类图具有O(Δ/log(Δ/√k))-可着色性。对于任意α∈(0,1)且k≪Δ^(2α)，令G为k-局部稀疏图。当q=Θ(Δ/log(Δ^α/√k))时，我们证明每个顶点采样O(Δ^α+√log n)种颜色即足以从采样颜色中获得G的正常q-着色。令k=1可得到前述Alon与Assadi关于无三角形图的结果。我们证明的关键要素是关于所谓色度设置下对应着色的一个命题，该命题改进了Anderson、Kuchukova及作者本人的近期工作，并具有独立的理论价值。