Abstract
We look at the minimal size of a maximal matching in general, bipartite and d-regular random graphs. We prove that the ratio between the sizes of any two maximal matchings approaches one in dense random graphs and random bipartite graphs. Weaker bounds hold for sparse random graphs and random d-regular graphs. We also describe an algorithm that with high probability finds a matching of size strictly less than n / 2 in a cubic graph. The result is based on approximating the algorithm dynamics by a system of linear differential equations.
Supported by EPSRC grant GR/L/77089.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bollobás, B.: Random Graphs. Academic Press, London (1985)
Edmonds, J.: Paths, Trees and Flowers. Canadian Journal of Math. 15, 449–467 (1965)
Erdos, P., Rényi, A.: On the Existence of a Factor of Degree One of a Connected Random Graph. Acta Mathematica Academiae Scientiarum Hungaricae 17(3–4), 359–368 (1966)
Grimmett, G.R., McDiarmid, C.J.H.: On Colouring Random Graphs. Mathematical Proceedings of the Cambridge Philosophical Society 77, 313–324 (1975)
Hopcroft, J., Karp, R.: An \(({\it n}{^5/2})\) Algorithm for Maximal Matching in Bipartite Graphs. SIAM Journal on Computing 2, 225–231 (1973)
Korte, B., Hausmann, D.: An Analysis of the Greedy Heuristic for Independence Systems. Annals of Discrete Mathematics 2, 65–74 (1978)
McKay, B.D.: Independent Sets in Regular Graphs of High Girth. Ars Combinatoria 23A, 179–185 (1987)
Micali, S., Vazirani, V.V.: An O(v 1/2 e) Algorithm for Finding Maximum Matching in General Graphs. In: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, New York, pp. 17–27 (1980)
Wormald, N.C.: Differential Equations for Random Processes and Random Graphs. Annals of Applied Probability 5, 1217–1235 (1995)
Yannakakis, M., Gavril, F.: Edge Dominating Sets in Graphs. SIAM Journal on Applied Mathematics 38(3), 364–372 (1980)
Zito, M.: Randomised Techniques in Combinatorial Algorithmics. PhD thesis, Department of Computer Science, University of Warwick (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zito, M. (2000). Small Maximal Matchings in Random Graphs. In: Gonnet, G.H., Viola, A. (eds) LATIN 2000: Theoretical Informatics. LATIN 2000. Lecture Notes in Computer Science, vol 1776. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10719839_2
Download citation
DOI: https://doi.org/10.1007/10719839_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67306-4
Online ISBN: 978-3-540-46415-0
eBook Packages: Springer Book Archive