具有稀疏邻域的图的调色板稀疏化
Palette Sparsification for Graphs with Sparse Neighborhoods
AI总结：
本研究针对k-局部稀疏图，证明每顶点采样O(Δ^α+√log n)种颜色即可得到正常q-着色，改进了对应着色相关结果，可复现无三角形图的已有结论。
摘要：
Assadi、Chen和Khanna的一项开创性调色板稀疏化结果表明，在每一个最大度为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，几乎必然能从采样颜色中得到一个正常着色。Alon和Assadi拓展了这项工作，证明了无三角形图的O(Δ/log Δ)-着色也有类似结果。他们的结果不仅从组合角度来看很有价值，还在不同计算模型下的图着色算法设计中有多种应用。在本工作中，我们关注局部稀疏图，即邻域稀疏的图。我们称图G=(V,E)是k-局部稀疏的，若对每个顶点v∈V，子图G[N(v)]至多包含k条边。Alon、Krivelevich和Sudakov的一项著名结果表明，这类图可进行O(Δ/log(Δ/√k))-着色。对任意α∈(0,1)且k≪Δ^(2α)，设G是一个k-局部稀疏图。当q=Θ(Δ/log(Δ^α/√k))时，我们证明为每个顶点采样O(Δ^α + √log n)种颜色足以从采样颜色中得到G的正常q-着色。令k=1可得到前述Alon和Assadi关于无三角形图的结果。我们证明中的一个关键要素是关于所谓色度设置下对应着色的一个命题，它改进了Anderson、Kuchukova和作者近期的工作，且本身也具有独立研究价值。
英文摘要：
A seminal palette sparsification result of Assadi, Chen, and Khanna states that in every $n$-vertex graph of maximum degree $Δ$, sampling $Θ(\log n)$ colors per vertex from $\{1, \ldots, Δ+1\}$ almost certainly allows for a proper coloring from the sampled colors. Alon and Assadi extended this work proving a similar result for $O\left(Δ/\log Δ\right)$-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 $O(Δ/\log (Δ/\sqrt{k}))$-colorable. For any $α\in (0, 1)$ and $k \ll Δ^{2α}$, let $G$ be a $k$-locally-sparse graph. For $q = Θ\left(Δ/\log \left(Δ^α/\sqrt{k}\right)\right)$, we show that sampling $O\left(Δ^α+ \sqrt{\log n}\right)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
