Linial’s famous color reduction algorithm reduces a given m-coloring of a graph with maximum degree \(\varDelta \) to an \(O(\varDelta ^2\log m)\)-coloring, in a single round in the LOCAL model. We give a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m-coloring in a directed graph of maximum outdegree \(\beta \), if every node has a list of size \(\varOmega (\beta ^2 (\log \beta +\log \log m + \log \log |{\mathcal {C}}|))\) from a color space \({\mathcal {C}}\) then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial’s color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local \(({\text {deg}}+1)\)-list coloring algorithm from Barenboim et al. (PODC, pp 437–446, 2018) by slightly reducing the runtime to \(O(\sqrt{\varDelta \log \varDelta })+\log ^* n\) and significantly reducing the message size (from \(\varDelta ^{O(\log ^* \varDelta )}\) to roughly \(\varDelta \)). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. (in: FOCS, pp 625–634, 2016).

Linial 著名的减色算法在 LOCAL 模型中的单轮中将具有最大度数\(\varDelta \)的图的给定m着色减少为\(O(\varDelta ^2\log m)\) -着色. 当节点被限制从允许的颜色列表中选择它们的颜色时，我们给出了类似的结果：如果每个节点都有一个大小为\( \来自颜色空间\({\mathcal {C}}\)的varOmega (\beta ^2 (\log \beta +\log \log m + \log \log |{\mathcal {C}}|))\ )然后他们可以在 LOCAL 模型中分两轮选择一种颜色。此外，节点的通信本质上包括将其列表发送给邻居。这是作为框架的一部分获得的，该框架还包含作为特例的 Linial 的颜色减少（带有替代证明）。我们的结果也导致了一个有缺陷的列表着色算法。作为推论，我们改进了来自 Barenboim 等人的最先进的真正本地 \(({\text {deg}}+1)\) -list 着色算法。(PODC, pp 437–446, 2018) 通过将运行时间略微减少到\(O(\sqrt{\varDelta \log \varDelta })+\log ^* n\)并显着减少消息大小（从\(\ varDelta ^{O(\log ^* \varDelta )}\)到大致\(\varDelta \)）。我们的技术受到Fraigniaud 等人的局部冲突着色框架的启发。（在：FOCS，第 625-634 页，2016 年）。