Given a graph $G$, an edge-coloring is an assignment of colors to edges of $G$ such that any two edges sharing an endpoint receive different colors. By Vizing's celebrated theorem, any graph of maximum degree $\Delta$ needs at least $\Delta$ and at most $(\Delta + 1)$ colors to be properly edge colored. In this paper, we study edge colorings in the streaming setting. The edges arrive one by one in an arbitrary order. The algorithm takes a single pass over the input and must output a solution using a much smaller space than the input size. Since the output of edge coloring is as large as its input, the assigned colors should also be reported in a streaming fashion. The streaming edge coloring problem has been studied in a series of works over the past few years. The main challenge is that the algorithm cannot "remember" all the color assignments that it returns. To ensure the validity of the solution, existing algorithms use many more colors than Vizing's bound. Namely, in $n$-vertex graphs, the state-of-the-art algorithm with $\widetilde{O}(n s)$ space requires $O(\Delta^2/s + \Delta)$ colors. Note, in particular, that for an asymptotically optimal $O(\Delta)$ coloring, this algorithm requires $\Omega(n\Delta)$ space which is as large as the input. Whether such a coloring can be achieved with sublinear space has been left open. In this paper, we answer this question in the affirmative. We present a randomized algorithm that returns an asymptotically optimal $O(\Delta)$ edge coloring using $\widetilde{O}(n \sqrt{\Delta})$ space. More generally, our algorithm returns a proper $O(\Delta^{1.5}/s + \Delta)$ edge coloring with $\widetilde{O}(n s)$ space, improving prior algorithms for the whole range of $s$.

翻译：给定图$G$，边染色是为$G$的每条边分配一种颜色，使得共享端点的任意两条边颜色不同。根据Vizing著名定理，最大度为$\Delta$的图至少需要$\Delta$种颜色，至多需要$(\Delta+1)$种颜色才能完成正常边染色。本文研究流式场景下的边染色问题。边按任意顺序逐条到达，算法对输入进行单趟扫描，并需使用远小于输入规模的空间输出解。由于边染色的输出规模与输入相当，分配的颜色也应以流式方式报告。近年来，流式边染色问题在一系列研究中得到探讨。核心挑战在于算法无法"记住"其返回的所有颜色分配。为确保解的合法性，现有算法使用的颜色数远超Vizing界。具体而言，在$n$顶点图中，使用$\widetilde{O}(n s)$空间的最优算法需要$O(\Delta^2/s + \Delta)$种颜色。特别值得注意的是，对于渐近最优的$O(\Delta)$染色，该算法需要$\Omega(n\Delta)$空间——这与输入规模相当。此类染色能否以亚线性空间实现一直是未解难题。本文对此给出肯定回答。我们提出一种随机算法，使用$\widetilde{O}(n \sqrt{\Delta})$空间即可返回渐近最优的$O(\Delta)$边染色。更一般地，我们的算法能以$\widetilde{O}(n s)$空间返回正常的$O(\Delta^{1.5}/s + \Delta)$边染色，在$s$的整个取值范围内优于先前算法。