Every graph with maximum degree $\Delta$ can be colored with $(\Delta+1)$ colors using a simple greedy algorithm. Remarkably, recent work has shown that one can find such a coloring even in the semi-streaming model. But, in reality, one almost never needs $(\Delta+1)$ colors to properly color a graph. Indeed, the celebrated \Brooks' theorem states that every (connected) graph beside cliques and odd cycles can be colored with $\Delta$ colors. Can we find a $\Delta$-coloring in the semi-streaming model as well? We settle this key question in the affirmative by designing a randomized semi-streaming algorithm that given any graph, with high probability, either correctly declares that the graph is not $\Delta$-colorable or outputs a $\Delta$-coloring of the graph. The proof of this result starts with a detour. We first (provably) identify the extent to which the previous approaches for streaming coloring fail for $\Delta$-coloring: for instance, all these approaches can handle streams with repeated edges and they can run in $o(n^2)$ time -- we prove that neither of these tasks is possible for $\Delta$-coloring. These impossibility results however pinpoint exactly what is missing from prior approaches when it comes to $\Delta$-coloring. We then build on these insights to design a semi-streaming algorithm that uses $(i)$ a novel sparse-recovery approach based on sparse-dense decompositions to (partially) recover the "problematic" subgraphs of the input -- the ones that form the basis of our impossibility results -- and $(ii)$ a new coloring approach for these subgraphs that allows for recoloring of other vertices in a controlled way without relying on local explorations or finding "augmenting paths" that are generally impossible for semi-streaming algorithms. We believe both these techniques can be of independent interest.

翻译：每个最大度为$\Delta$的图均可通过简单贪心算法使用$(\Delta+1)$种颜色完成着色。值得注意的是，近期研究表明即便在半流模型下也能实现此类着色。然而现实中几乎无需使用$(\Delta+1)$种颜色对图进行正常着色——事实上，著名的Brooks定理指出：除团和奇圈外的所有（连通）图均可使用$\Delta$种颜色着色。那么，我们是否也能在半流模型下找到$\Delta$着色方案？本文通过设计随机化半流算法肯定了该核心问题：对于任意输入图，该算法能以高概率正确判定其是否可$\Delta$着色，或输出该图的$\Delta$着色方案。该结果的证明始于一个迂回策略。我们首先（可证明地）揭示了先前流式着色方法在$\Delta$着色问题上的失效边界：例如，所有先前方法虽能处理包含重复边的流并可在$o(n^2)$时间内运行，但我们证明这两项任务对$\Delta$着色均不可行。这些不可能性结果恰好指明了先前方法在$\Delta$着色中的缺失环节。基于这些洞见，我们设计了半流算法，其采用：(i) 基于稀疏-稠密分解的新型稀疏恢复方法，用以（部分）恢复输入图中的"问题子图"——即构成我们不可能性结果基础的结构；(ii) 针对这些子图的新着色方法，可在不依赖局部探索或寻找"增广路径"（这对半流算法通常不可行）的情况下，以受控方式对其他顶点进行重着色。我们相信这两种技术均具有独立研究价值。