Faster Deterministic Streaming Vertex Coloring

Graph coloring is a fundamental problem in computer science. In the semi-streaming model, an input graph $G$ on $n$ vertices and maximum degree $\Delta$ is presented as a stream of edges, and the goal is to compute a vertex coloring using a small number of colors while storing only $\tilde{O}(n)$ bits of memory. Recent work has revealed an exponential separation between randomized and deterministic approaches in this setting: while randomized algorithms can achieve a $(\Delta+1)$-coloring in a single pass [Assadi, Chen, and Khanna, 2019], any single-pass deterministic algorithm requires $\exp(\Delta^{\Omega(1)})$ colors [Assadi, Chen, and Sun, 2022]. Consequently, deterministic algorithms that use few colors must necessarily make multiple passes over the stream. Prior to this work, the best known deterministic trade-offs were: an $O(\Delta^2)$-coloring in 2 passes, an $O(\Delta)$-coloring in $O(\log \Delta)$ passes [Assadi, Chen, and Sun, 2022], and a $(\Delta+1)$-coloring in $O(\log \Delta \cdot \log\log \Delta)$ passes [Assadi, Chakrabarti, Ghosh, and Stoeckl, 2023]. It remained open whether better trade-offs -- particularly with sub-logarithmic pass complexity and linear-in-$\Delta$ palette size -- were achievable. In this paper, we present a new deterministic semi-streaming algorithm that computes an $O(\Delta)$-coloring in $O(\sqrt{\log \Delta})$ passes. This is the first deterministic streaming algorithm to achieve a coloring with palette size linear-in-$\Delta$ using sublogarithmic-in-$\Delta$ passes.