Density-Sensitive Algorithms for (∆ + 1)-Edge Coloring

Vizing's theorem asserts the existence of a (∆ + 1)-edge coloring for any graph G, where ∆ = ∆(G) denotes the maximum degree of G. Several polynomial time (∆ + 1)-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is Õ(min{m^√n,m∆}),¹ by Gabow, Nishizeki, Kariv, Leven and Terada from 1985, where n and m denote the number of vertices and edges in the graph, respectively. Recently, Sinnamon shaved off a polylog(n) 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 ≤ ∆ in any graph, many natural and real-world graphs exhibit a significant separation between α and ∆. In this work we design a (∆ + 1)-edge coloring algorithm with a running time of Õ(min{m^√n,m∆})· _∆^α, thus improving the longstanding time barrier by a factor of _∆^α. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., α = Õ(1)) as well as when α = Õ(^√∆_n). Our algorithm builds on Gabow et al.'s and Sinnamon's algorithms, and can be viewed as a density-sensitive refinement of them.