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