An Algorithm for Computing Edge Colorings on Regular Bipartite Multigraphs

In this paper, we consider the problem of finding an edge coloring of a d-regular bipartite multigraph with 2n vertices and m = nd edges. The best known deterministic algorithm (by Cole, Ost, and Schirra) takes O(m log d) time to find an edge coloring of such a graph. This bound is achieved by combining an O(m)-time perfect-matching algorithm with the Euler partition algorithm. The O(m) time bound on the Cole, Ost, and Schirra perfect-matching algorithm has been shown to be optimal. In this paper we present an alternative perfect-matching algorithm called QuickMatch. Empirical analysis shows that QuickMatch finds a perfect matching in O(m) time in the average case. The QuickMatch algorithm allows us to compute an edge coloring in O(m log d) time in the average case. Due to its simplicity, the presented method is easy to implement and the constants in the time bound are small. Because of these features, QuickMatch is a highly practical and competitive method for finding edge colorings of d-regular bipartite multigraphs.