Edge-Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges, and maximum degree Δ, we find an edge-coloring for G using Δ colors in time T+O(mlogΔ), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k≤Δ. Together with best known bounds for T this implies on O(mlogΔ+mΔlogmΔlog2Δ) edge-coloring algorithm which improves on the O(mlogΔ+mΔlogmΔlog3Δ) algorithm of Hopcroft and Cole. Our algorithm can also be used to find a (Δ+2)-edge-coloring for G in time O(mlogΔ). The previous best approximation algorithm with the same time bound needed Δ+logΔ colors.