Parallel Algorithms for the Edge-Coloring and Edge-Coloring Update Problems

with D 1 1 colors for the edge-coloring problem. However, Holyer has shown that deciding whether a graph requires D or D 1 1 colors is NP-complete [10]. For a multigraph G, Shannon showed that x9(G) # 3D/2 [16]. A number of parallel algorithms exist for the edge-coloring problem. Lev et al. [14] presented parallel edge-coloring algorithms with D colors for bipartite multigraphs. When D 5 2, their algorithm requires O(log D log n) time and O(nD) processors. Otherwise, their algorithm requires O(log D log2 n) time and O(nD) processors on the EREW PRAM. Gibbons et al. [5] and Gibbons and Rytter [6] suggested algorithms for some other special graphs such as trees, outerplanar graphs, and Halin graphs. For trees, their algorithm requires O(log n) time and O(n) processors; for outplanar graphs, their algorithm requires O(log3 n) time and O(n2) processors; for Halin graphs, their algorithm requires O(log2 n) time and O(n) processors. All of their algorithms run in the CREW PRAM, and the number of colors used is at most D. For planar graphs, Chrobak and Yung [3] presented an edge-coloring algorithm with maxhD, 19j colors. Their algorithm runs in O(log2 n) time and uses O(n) processors on the EREW PRAM [3]. Later, Chrobak and Nishizeki [4] improved the algorithm in [3] by reducing the number of colors to maxhD, 8j. Their algorithm requires O(log3 n) time and O(n2) processors [4]. He [9] also discussed the edge-coloring problem with D colors for planar graphs, his algorithm has the same complexity as that of [3]. For general graphs, the pioneer work was done by Karloff and Shmoys [13]. They presented an edge-coloring algorithm with D 1 1 colors for simple graphs, which requires O(D6 log4 n) time and O(n2D) processors, assuming the fastest known algorithm for finding maximal independent sets [7] is used. They also gave a randomized edge-coloring algorithm with D 1 20D1/21« colors, (« # 1/4), which runs in O(logO(1) n) expected time and uses O(nO(1)) processors (independent of «). For multigraphs, Upfal once presented an O(log3 Dn) time algorithm with 3D/2 colors by using O(Dn) processors (appears in [13]). For the special class of multigraphs of D 5 3, Karloff and Shmoys [13] presented an algorithm which runs in O(log n) time and uses O(n) processors. In this paper we study two problems. The first problem JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 32, 66–73 (1996) ARTICLE NO. 0005