Density-Sensitive Algorithms for (Δ+ 1)-Edge Coloring
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Vizing's theorem asserts the existence of a (Δ+1)-edge coloring for any graph G, where Δ = Δ(G) denotes the maximum degree of G. Several polynomial time (Δ+1)-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is Õ(min{m ·√(n), m ·Δ}), by Gabow et al. from 1985, where n and m denote the number of vertices and edges in the graph, respectively. (The Õ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The arboricity α = α(G) of a graph G is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's “uniform density”. While α≤Δ in any graph, many natural and real-world graphs exhibit a significant separation between α and Δ. In this work we design a (Δ+1)-edge coloring algorithm with a running time of Õ(min{m ·√(n), m ·Δ})·α/Δ, thus improving the longstanding time barrier by a factor of α/Δ. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., α = Õ(1)) as well as when α = Õ(Δ/√(n)). Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.
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