Edge­Coloring Bipartite Graphs

Given a bipartite graph G with n nodes, m edges and maximum degree D, we find an edge coloring for G using D colors in time T + O(m log D), where T is the time needed to find a perfect matching in a k-regular bipartite graph with O(m) edges and k <= D. Together with best known bounds for T, this implies an O(m log D + (m/D) log (m/D) log^2 D) edge-coloring algorithm which improves on the O(m log D + (m/D) log (m/D) log^3 D) algorithm of Hopcroft and Cole. Our algorithm can also be used to find a (D+2)-edge-coloring for G in time O(m log D). The previous best approximation algorithm with the same time bound needed D + log D colors.