Reed [J. Comb. Theory B, 1999] showed that graphs of maximum degree Δ ⩾ 10^14 without Δ-cliques are (Δ-1)-colorable. We design a one-pass semi-streaming algorithm for computing such a coloring. Additionally, we prove that any one-pass (Δ-k)-coloring algorithm for 0 ⩽ k < (Δ+1)/2 requires Ω(n(k+1)) space.

We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.

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We consider the problem of maintaining a proper $(Δ+ 1)$-vertex coloring in a graph on $n$-vertices and maximum degree $Δ$ undergoing edge insertions and deletions. We give a randomized algorithm with amortized update time $\widetilde{O}( n^{2/3} )$ against adaptive adversaries, meaning that updates may depend on past decisions by the algorithm. This improves on the very recent $\widetilde{O}( n^{8/9} )$-update-time algorithm by Behnezhad, Rajaraman, and Wasim (SODA 2025) and matches a natural barrier for dynamic $(Δ+1)$-coloring algorithms. The main improvements are in the densest regions of the graph, where we use structural hints from the study of distributed graph algorithms.

Graph coloring is fundamental to distributed computing. We give the first sub-logarithmic distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges, and they appear frequently as components in the study of distributed algorithms. In particular, we give a O (log* n)-round algorithm to (Δ + 1)-color cluster graphs of at least polylogarithmic degree. The previous best bound known was poly(log n) [Flin et al., SODA'24]. This properly generalizes results in the CONGEST model and shows that distributed graph problems can be solved quickly even when the node itself is decentralized.

We study the communication complexity of (Δ + 1) vertex coloring, where the edges of an n-vertex graph of maximum degree Δ are partitioned between two players. We provide a randomized protocol which uses O(n) bits of communication and ends with both players knowing the coloring. Combining this with a folklore Ω(n) lower bound, this settles the randomized communication complexity of (Δ + 1)-coloring up to constant factors.

The celebrated palette sparsification result of [Assadi, Chen, and Khanna SODA’19] shows that to compute a Δ + 1 coloring of the graph, where Δ denotes the maximum degree, it suffices if each node limits its color choice to O(log n) independently sampled colors in {1, 2,…, Δ + 1}. They showed that it is possible to color the resulting sparsified graph—the spanning subgraph with edges between neighbors that sampled a common color, which are only Õ(n) edges—and obtain a Δ + 1 coloring for the original graph. However, to compute the actual coloring, that information must be gathered at a single location for centralized processing. We seek instead a local algorithm to compute such a coloring in the sparsified graph. The question is if this can be achieved in poly (log n) distributed rounds with small messages.

Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph $H$ to be colored is locally embedded within the communication graph $G$. Besides generalizing classical distributed graph coloring (where $H=G$), this captures other previously studied settings, including cluster graphs and power graphs. We find that the complexity of coloring a virtual graph depends on the edge congestion of its embedding. The main question of interest is how fast we can color virtual graphs of constant congestion. We find that, surprisingly, these graphs can be colored nearly as fast as ordinary graphs. Namely, we give a $O(\log^4\log n)$-round algorithm for the deg+1-coloring problem, where each node is assigned more colors than its degree. This can be viewed as a case where a distributed graph problem can be solved even when the operation of each node is decentralized.

We present an O(log3 log n)-round distributed algorithm for the (Δ + 1)-coloring problem, where each node broadcasts only one O(log n)-bit message per round to its neighbors. Previously, the best such broadcast-based algorithm required O(log n) rounds. If Δ ∈ Ω(log 3 n), our algorithm runs in O(log* n) rounds. Our algorithm's round complexity matches the state-of-the-art in the much more powerful CONGEST model [Halldórsson et al., STOC'21 & PODC'22], where each node sends one different message to each of its neighbors, thus sending up to Θ(n log n) bits per round. This is the best complexity known, even if message sizes are unbounded.

We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2 coloring in CONGEST with $Δ^2+1$ colors. For $Δ\gg\operatorname{poly}\log n$, this improves exponentially on the $O(\logΔ+\operatorname{poly}\log\log n)$ algorithm of [Halldórsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lovász Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.