# Parallel algorithms for approximate edge colouring of simple graphs

Session 4 Parallel Algorithms

First Online:

## Abstract

Two parallel algorithms for edge-colouring simple graphs are presented. One takes *O*(*m*log*n*) time using a polynomial number of processors on an SIMD parallel computer which allows read conflicts but no write conflicts. The second algorithm uses the first in a divide-and-conquer setting and takes *O*(*n*log^{2}*n*) time at the cost of a factor of *n* extra processors on the same model of computation. How to obtain improved time bounds from these algorithms for some special types of graph is also discussed.

Either algorithm uses no more than φ_{e}+1 colours where φ_{e} is the edge-chromatic number of the graph being coloured. Moreover the expected performance of each of the algorithms is optimal.

## Keywords

edge-colouring graphs algorithm: approximation probabilistic parallel SIMD computer## Preview

Unable to display preview. Download preview PDF.

## References

- [1]M. Atallah and U. Vishkin, "Finding Euler Tours in Parallel",
*J. Comput. and Syst. Sciences***29**, 330–337 (1984).CrossRefGoogle Scholar - [2]R.D. Dutton and R.C. Brigham, "A new graph colouring algorithm",
*Computer J.***24**, 85–86 (1981).CrossRefGoogle Scholar - [3]A.M. Frieze, "Parallel Algorithms for Finding Hamiltonian Cycles in Random Graphs",
*Manuscript*, Department of Computer Science, Queen Mary College (March 1986).Google Scholar - [4]M.R. Garey and D.S. Johnson,
*"Computer and Intractability: A Guide to the Theory of NP-completeness"*, Freeman (1979).Google Scholar - [5]H.N. Grabow and O. Kariv, "Algorithms for Edge Colouring Bipartite Graphs and Multigraphs",
*SIAM J. Comput.***11**, 117–129 (1982).CrossRefGoogle Scholar - [6]A.M. Gibbons,
*"Algorithmic Graph Theory"*, Cambridge University Press (1985).Google Scholar - [7]A.M. Gibbons and O.A. Ogunyode, "A Polynomial-Time Algorithm to Edge-Colour Almost All Graphs Using φ
_{e}Colours",*Theory of Computation, Report No.***68**, (September 1984).Google Scholar - [8]I. Holyer, "The NP-completeness of Edge-Colouring",
*SIAM J. Comput.***10**, 718–720 (1981).CrossRefGoogle Scholar - [9]F.T. Leighton, "A graph coloring algorithm for large scheduling problems",
*J. Res. Natn. Bur. Stand.***84**, 489–506 (1979).Google Scholar - [10]G. Lev, N. Pippenger and L.G. Valliant, "A Fast Parallel Algorithm for Routing in Permutation Networks",
*IEEE Trans. Comput.*, C-30, 93–110 (1981).Google Scholar - [11]O.A. Ogunyode, "Approximation and Parallel Algorithms for Some
*NP*-Hard Problems",*Ph.D. Thesis*, Department of Computer Science, University of Warwick (August 1986).Google Scholar - [12]M.J. Quinn and N. Deo, "A Parallel Approximate Algorithm for the Euclidean Traveling Salesman Problem",
*Report CS-83-105*, Computer Science Department, Washington State University, Pullman (1983).Google Scholar - [13]Y.H. Tsin and F.Y. Chin, "Efficient Parallel Algorithms for a Class of Graph Theoretic Problems",
*SIAM J. Comput.***13**, 580–599 (1984).CrossRefGoogle Scholar - [14]V.G. Vizing, "On an estimate of the chromatic class of p-graph",
*Diskret. Analiz.***3**, 25–30 (1964).Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1987