Brooks' Theorem in Graph Streams: A Single-Pass Semi-Streaming Algorithm for $Δ$-Coloring

TL;DR

This work resolves the question of Δ-coloring in the semi-streaming model by giving a randomized single-pass algorithm that either certifies non-Δ-colorability or outputs a Δ-coloring for connected graphs that are not cliques or odd cycles. The approach combines a refined palette sparsification framework with a sparse-dense decomposition to identify and implicitly recover problematic subgraphs, followed by a recoloring-based coloring procedure that uses helper structures to extend partial colorings without reliance on augmenting paths. The key technical contributions are (i) a detailed understanding of why palette sparsification alone fails for Δ-coloring, (ii) a sparse-recovery mechanism that discovers and partially recovers critical subgraphs via sparse-dense decomposition, and (iii) a final six-phase coloring procedure that recolors and extends colorings through helper structures, achieving a Δ-coloring in the semi-streaming model. The results advance our understanding of graph coloring in restricted computational models and have potential implications for related streaming and distributed coloring problems, including dynamic streams and MPC.

Abstract

Every graph with maximum degree $Δ$ can be colored with $(Δ+1)$ colors using a simple greedy algorithm. Remarkably, recent work has shown that one can find such a coloring even in the semi-streaming model. But, in reality, one almost never needs $(Δ+1)$ colors to properly color a graph. Indeed, the celebrated \Brooks' theorem states that every (connected) graph beside cliques and odd cycles can be colored with $Δ$ colors. Can we find a $Δ$-coloring in the semi-streaming model as well? We settle this key question in the affirmative by designing a randomized semi-streaming algorithm that given any graph, with high probability, either correctly declares that the graph is not $Δ$-colorable or outputs a $Δ$-coloring of the graph. The proof of this result starts with a detour. We first (provably) identify the extent to which the previous approaches for streaming coloring fail for $Δ$-coloring: for instance, all these approaches can handle streams with repeated edges and they can run in $o(n^2)$ time -- we prove that neither of these tasks is possible for $Δ$-coloring. These impossibility results however pinpoint exactly what is missing from prior approaches when it comes to $Δ$-coloring. We then build on these insights to design a semi-streaming algorithm that uses $(i)$ a novel sparse-recovery approach based on sparse-dense decompositions to (partially) recover the "problematic" subgraphs of the input -- the ones that form the basis of our impossibility results -- and $(ii)$ a new coloring approach for these subgraphs that allows for recoloring of other vertices in a controlled way without relying on local explorations or finding "augmenting paths" that are generally impossible for semi-streaming algorithms. We believe both these techniques can be of independent interest.

Figures (6)

• Figure 4: A graph with maximum degree $\Delta=4$ and its sparse-dense decomposition in (a) (each box denotes an almost-clique and remaining vertices are sparse). Part (b) is an illustration of why palette sparsification fails for $\Delta$-coloring: the only way to $\Delta$-color this graph is to color the marked vertices the same, which cannot be done with these sampled lists. Part (c) shows a similar construction can be used to prove a query lower bound for $\Delta$-coloring. (The actual instance is obtained from $\Theta(n/\Delta)$ copies of such pairs.)
• Figure 5: An illustration of three possible types of outer connections on a graph with maximum degree $\Delta=6$. The almost-clique in part (a) has a "right" type of outside connection and is going to receive a more "random" coloring on its neighbors, compared to the almost-clique in part (b) with "few" outside neighbors and part (c) with "too many" ones. In particular, the latter almost-cliques now cannot be $\Delta$-colored without changing the color of outside vertices as the same colors are blocked for all vertices of the inner (actual) cliques.
• Figure 6: An illustration of sparse-recovery on the neighborhood of each vertex, plus an algorithm that finds the identity of vertices in each almost-clique, allows for recovering all edges "highly-dense" almost-cliques. Our actual algorithm is considerably more involved as it needs to partially recover "not-too-dense" almost-cliques also.
• Figure 7: Two problematic almost-cliques in a graph with maximum degree $\Delta=4$. Both almost-cliques are hard for palette sparsification. The only way part (a) can be $\Delta$-colored is if the vertices incident on the non-edge have intersecting lists. Part (b) is not$\Delta$-colorable if the outside (marked) vertices are all colored the same.
• Figure 8: An illustration of base palette graphs (\ref{['def:palette-graph']}) and sampled palette graphs (\ref{['def:sampled-palette-graph']}).

Theorems & Definitions (52)

• Theorem 1.1: Semi-Streaming Brooks' Theorem
• proposition 3.2: Chernoff bound; c.f. DubhashiP09
• definition 3.3
• definition 3.4
• proposition 3.5: Sparse-Dense Decomposition; cf. MolloyR13AssadiCK19aAssadiC22
• proposition 3.6: cf. DasV13
• proposition 3.7
• definition 3.8
• definition 3.9
• Lemma 3.10: AssadiCK19a