Edge Coloring Series Parallel Graphs

Abstract

We present an algorithm for optimally edge coloring series parallel graphs. We give a linear time implementation, as well as a parallel implementation, of the algorithm that runs in O(log 3 n) time using O( n) processors. The sequential implementation, which is optimal, improves the best-known algorithm. The parallel implementation of the algorithm is the first known NC algorithm for this problem. The algorithm is based on the ear decomposition of a graph (Y. Maon, B. Schieber, and U. Vishkin, Parallel ear decomposition search (EDS) and ST-Numbering in graphs, Theoret. Comput. Sci. 47 (1986), 277-298). Eppstein (Parallel recognition of series-parallel graphs, Inform. Comput. 98 (1992), 41-55) found that any ear decomposition of a series parallel graph is nested. We show constructively that for every biconnected series parallel graph there exists an open ear decomposition, such that its corresponding tree of ears has an O(log n) depth, and this ear decomposition contains no ear whose endpoints are connected by a single edge in its parent. This result is used to reduce a match in series parallel graphs into a match in outerplanar graphs, and to establish the edge coloring problem of series parallel graphs in NC.