Faster Distributed Δ-Coloring via a Reduction to MIS

Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of O(log 5/3 n) rounds for maximal independent set (MIS) and (∆ + 1)-coloring [Ghaffari, Grunau, FOCS'24] and O(log 19/9 n) rounds for the more restrictive ∆-coloring problem [Ghaffari, Kuhn, FOCS'21; Ghaffari, Grunau, FOCS'24; Bourreau, Brandt, Nolin, STOC'25]. In our work, we show that ∆-coloring can be solved deterministically in O(log 5/3 n) rounds as well, matching the currently best bound for (∆ + 1)-coloring.

We achieve our result by developing a reduction from ∆-coloring to MIS that guarantees that the (asymptotic) complexity of ∆-coloring is at most the complexity of MIS, unless MIS can be solved in sublogarithmic time, in which case, due to the Ω(log n)-round ∆-coloring lower bound from [BFHKLRSU, STOC'16], our reduction implies a tight complexity of Θ(log n) for ∆-coloring. In particular, any improvement on the complexity of the MIS problem will yield the same improvement for the complexity of ∆-coloring (up to the true complexity of ∆-coloring).

Our reduction also yields improvements for ∆-coloring in the randomized LOCAL model and when complexities are parameterized by both n and ∆. For instance, we obtain a randomized complexity bound of O(log 5/3 log n) rounds (improving over the state of the art of O(log 8/3 log n) rounds) on general graphs and tight complexities of Θ(log n) and Θ(log log n) for the deterministic, resp. randomized, complexity on bounded-degree graphs. In the special case of graphs of constant clique number (which for instance include bipartite graphs), we additionally give a reduction to the (∆ + 1)-coloring problem.