This paper investigates the semi-streaming complexity of \textit{$k$-partial coloring}, a generalization of proper graph coloring. For $k \geq 1$, a $k$-partial coloring requires that each vertex $v$ in an $n$-node graph is assigned a color such that at least $\min\{k, °(v)\}$ of its neighbors are assigned colors different from its own. This framework naturally extends classical coloring problems: specifically, $k$-partial $(k+1)$-coloring and $k$-partial $k$-coloring generalize $(Δ+1)$-proper coloring and $Δ$-proper coloring, respectively. Prior works of Assadi, Chen, and Khanna [SODA~2019] and Assadi, Kumar, and Mittal [TheoretiCS~2023] show that both $(Δ+1)$-proper coloring and $Δ$-proper coloring admit one-pass randomized semi-streaming algorithms. We explore whether these efficiency gains extend to their partial coloring generalizations and reveal a sharp computational threshold : while $k$-partial $(k+1)$-coloring admits a one-pass randomized semi-streaming algorithm, the $k$-partial $k$-coloring remains semi-streaming intractable, effectively demonstrating a ``dichotomy of one color'' in the streaming model.

翻译：本文研究了\textit{$k$-部分着色}的半流式计算复杂度，该问题是传统图着色的推广。对于$k \geq 1$，$k$-部分着色要求为$n$节点图中的每个顶点$v$分配一种颜色，使得其至少$\min\{k, °(v)\}$个邻接顶点被分配与其自身不同的颜色。该框架自然推广了经典着色问题：具体而言，$k$-部分$(k+1)$着色与$k$-部分$k$着色分别推广了$(Δ+1)$-正常着色与$Δ$-正常着色。Assadi、Chen和Khanna [SODA~2019] 以及Assadi、Kumar和Mittal [TheoretiCS~2023] 的前期工作表明，$(Δ+1)$-正常着色与$Δ$-正常着色均存在单轮随机化半流式算法。我们探究这些效率优势是否可推广至其部分着色变体，并揭示了一个尖锐的计算阈值：虽然$k$-部分$(k+1)$着色存在单轮随机化半流式算法，但$k$-部分$k$着色在流式计算模型中保持半流式不可解，这实质上证明了流式模型中“单色二分性”现象的存在。