Streaming Edge Coloring with Subquadratic Palette Size

In this paper, we study the problem of computing an edge-coloring in the (one-pass) W-streaming model. In this setting, the edges of an n-node graph arrive in an arbitrary order to a machine with a relatively small space, and the goal is to design an algorithm that outputs, as a stream, a proper coloring of the edges using the fewest possible number of colors. Behnezhad et al. [Behnezhad et al., 2019] devised the first non-trivial algorithm for this problem, which computes in Õ(n) space a proper O(∆²)-coloring w.h.p. (here ∆ is the maximum degree of the graph). Subsequent papers improved upon this result, where latest of them [Ansari et al., 2022] showed that it is possible to deterministically compute an O(∆²/s)-coloring in O(ns) space. However, none of the improvements succeeded in reducing the number of colors to O(∆^2−ϵ) while keeping the same space bound of Õ(n)¹. In particular, no progress was made on the question of whether computing an O(∆)-coloring is possible with roughly O(n) space, which was stated in [Behnezhad et al., 2019] to be an interesting open problem. In this paper we bypass the quadratic bound by presenting a new randomized Õ(n)-space algorithm that uses Õ(∆^1.5) colors.