Parallel algorithms for the edge-coloring and edge-coloring update problems

Let G(V, E) be a simple undirected graph with a maximum vertex degree Delta(G) (or Delta for short), /V/ = n and /E/ = m. An edge-coloring of G is an assignment to each edge in G a color such that all edges sharing a common vertex have different colors. The minimum number of colors needed is denoted by chi'(G) (called the chromatic index). For a simple graph G, it is known that Delta less than or equal to chi'(G) less than or equal to Delta + 1. This paper studies two edge-coloring problems. The first problem is to perform edge-coloring for an existing edge-colored graph G with Delta + 1 colors stemming from the addition of a new vertex into G. The proposed parallel algorithm for this problem runs in O(Delta(3/2)log(3) Delta + Delta log n) time using O(max(n Delta, Delta(3))) processors. The second problem is to color the edges of a given uncolored graph G with Delta + 1 colors. For this problem, our first parallel algorithm requires O(Delta(5.5)log(3) Delta log n + Delta(5) log(4) n) time and O(max{n(2) Delta, n Delta(3)}) processors, which is a slight improvement on the algorithm by H. J. Karloff and D. B. Shmoys [J. Algorithms 8 (1987), 39-52]. Their algorithm costs O(Delta(6) log(4) n) time and O(n(2) Delta) processors if we use the fastest known algorithm for finding maximal independent sets by M. Goldberg and T. Spencer [SIAM J. Discrete Math. 2 (1989), 322-328]. Our second algorithm requires O(Delta(4.5)log(3) Delta log n + Delta(4) log(4) n) time and O(max{n(2), n Delta(3)}) processors. Finally, we present our third algorithm by incorporating the second algorithm as a subroutine. This algorithm requires O(Delta(3.5)log(3) Delta log n + Delta(4) log(4) n) time and O(max{n(2) log Delta, n Delta(3)}) processors, which improves, by an O(Delta(2.5)) factor in time, on Karloff and Shmoys' algorithm. All of these algorithms run in the COMMON CRCW PRAM model. (C) 1996 Academic Press, Inc.