A Fast Algorithm for Computing a Nearly Equitable Edge Coloring with Balanced Conditions

We discuss the nearly equitable edge coloring problem on a multigraph and propose an ecient algorithm for solving the problem, which has a better time complexity than the previous algorithms. The coloring computed by our algorithm satises additional balanced conditions on the number of edges used in each color class, where conditions are imposed on the balance among all edges in the multigraph as well as the balance among parallel edges between each vertex pair. None of the previous algorithms are guaranteed to satisfy these balanced conditions simultaneously. To achieve these improvements, we propose a new recoloring procedure, which is based on a set of edge-disjoint alternating walks, while the existing algorithms are based on an Eulerian circuit or a single alternating walk. This new recoloring procedure makes it possible to reduce the time complexity of the algorithm.