Bipartite Edge Coloring in $O(\Delta m)$ Time

Abstract

We show that a minimum edge coloring of a bipartite graph can be found in $O(\Delta m)$ time, where $\Delta$ and m denote the maximum degree and the number of edges of G, respectively. It is equivalent to finding a perfect matching in a k-regular bipartite graph in O(km) time. By sharpening the methods, a minimum edge coloring of a bipartite graph can be found in $O((p_{\max}(\Delta)+\log \Delta)m)$ time, where $p_{\max}(\Delta)$ is the largest prime factor of $\Delta$. Moreover, a perfect matching in a k-regular bipartite graph can be found in O(pmax(k)m)time.