Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T23:06:42.932Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Correlation Inequality of Farr

Published online by Cambridge University Press:  12 September 2008

Colin McDiarmid
Colin McDiarmid
Affiliation:
Department of Statistics, Oxford University
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose that each vertex of a graph independently chooses a colour uniformly from the set {1, …, k}; and let Si be the random set of vertices coloured i. Farr shows that the probability that each set Si is stable (so that the colouring is proper) is at most the product of the k probabilities that the sets Si separately are stable. We give here a simple proof of an extension of this result.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 2 , June 1992 , pp. 157 - 160
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ahlswede, R. and Daykin, D. E. (1978) An inequality for the weights of two families of sets, their unions and intersections, Z. Wahrlichkeitstheorie verw. Gebeite 43 183–185.CrossRefGoogle Scholar
[2]Bollobás, B. (1986) Combinatorics, set systems, hypergraphs, families of vectors and combinatorial probability, Cambridge University Press.Google Scholar
[3]Farr, G. E. (to appear) A correlation inequality involving stable set and chromatic polynomials, J. Combinatorial Theory B.Google Scholar
[4]Graham, R. L. (1983) Applications of the FKG inequality and its Relatives, Mathematical Programming The State of the Art, Springer-Verlag, 115–131.CrossRefGoogle Scholar
[5]Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process, Proc. Camb. Phil. Soc. 56, 13–20.CrossRefGoogle Scholar
[6]McDiarmid, C. J. H. (1980) Clutter percolation and random graphs, Mathematical Programming Study 13, 17–25.CrossRefGoogle Scholar
[7]McDiarmid, C. J. H. (1981) General percolation and random graphs, Advances in Applied Probability 13, 40–60.CrossRefGoogle Scholar
[8]McDiarmid, C. J. H. (1987) Some inequalities for random graphs and percolation, Notes from New York Academy of Science Graph Theory Day 13, 9–13.Google Scholar