We study graph coloring problems in the streaming model, where the goal is to process an $n$-vertex graph whose edges arrive in a stream, using a limited space that is smaller than the trivial $O(n^2)$ bound. While prior work has largely focused on coloring graphs with a large number of colors, we explore the opposite end of the spectrum: deciding whether the input graph can be colored using only a few, say, a constant number of colors. We are interested in each of the adversarial, random order, or dynamic streams. Our work lays the foundation for this new direction by establishing upper and lower bounds on space complexity of key variants of the problem. Some of our main results include: - Adversarial: for distinguishing between $q$- vs $2^{\Omega(q)}$-colorable graphs, lower bounds of $n^{2-o(1)}$ space for $q$ up to $(\log{n})^{1/2-o(1)}$, and $n^{1+\Omega(1/\log\log{n})}$ space for $q$ further up to $(\log{n})^{1-o(1)}$. - Random order: for distinguishing between $q$- vs $q^t$-colorable graphs for $q,t \geq 2$, an upper bound of $\tilde{O}(n^{1+1/t})$ space. Specifically, distinguishing between $q$-colorable graphs vs ones that are not even poly$(q)$-colorable can be done in $n^{1+o(1)}$ space unlike in adversarial streams. Although, distinguishing between $q$-colorable vs $\Omega(q^2)$-colorable graphs requires $\Omega(n^2)$ space even in random order streams for constant $q$. - Dynamic: for distinguishing between $q$- vs $q \cdot t$-colorable graphs for any $q \geq 3$ and $t \geq 1$, nearly optimal upper and lower bounds of $\tilde{\Theta}(n^2/t^2)$ space. We develop several new technical tools along the way: cluster packing graphs, a generalization of Ruzsa-Szemer\'edi graphs; a player elimination framework based on cluster packing graphs; and new edge and vertex sampling lemmas tailored to graph coloring.

翻译：我们在流式模型中研究图着色问题，其目标是处理一个边以流形式到达的$n$顶点图，同时使用小于平凡$O(n^2)$界限的有限空间。以往的研究主要集中在使用大量颜色对图进行着色，而我们则探索了光谱的另一端：判定输入图是否仅能使用少量（例如常数个）颜色进行着色。我们对对抗流、随机顺序流或动态流中的每一种情况都感兴趣。我们的工作通过建立该问题关键变体的空间复杂度上界和下界，为这一新方向奠定了基础。我们的一些主要结果包括：- 对抗流：对于区分$q$可着色图与$2^{\Omega(q)}$可着色图，当$q$达到$(\log{n})^{1/2-o(1)}$时，空间下界为$n^{2-o(1)}$；当$q$进一步达到$(\log{n})^{1-o(1)}$时，空间下界为$n^{1+\Omega(1/\log\log{n})}$。- 随机顺序流：对于区分$q$可着色图与$q^t$可着色图（其中$q,t \geq 2$），空间上界为$\tilde{O}(n^{1+1/t})$。具体而言，与对抗流不同，区分$q$可着色图与甚至非多项式$(q)$可着色图可以在$n^{1+o(1)}$空间内完成。然而，即使对于常数$q$，在随机顺序流中区分$q$可着色图与$\Omega(q^2)$可着色图也需要$\Omega(n^2)$空间。- 动态流：对于区分$q$可着色图与$q \cdot t$可着色图（其中任意$q \geq 3$且$t \geq 1$），空间上界和下界几乎最优，为$\tilde{\Theta}(n^2/t^2)$。在此过程中，我们开发了几种新的技术工具：簇打包图（Ruzsa-Szemer\'edi图的推广）；基于簇打包图的玩家淘汰框架；以及专为图着色定制的新边和顶点采样引理。