Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-20T22:19:42.874Z Has data issue: false hasContentIssue false
  • English
  • Français

Achromatic numbers of random graphs

Published online by Cambridge University Press:  24 October 2008

Colin McDiarmid
Colin McDiarmid
Affiliation:
Wolfson College, Oxford
Get access Rights & Permissions [Opens in a new window]

Abstract

The achromatic number ψ(G) of a graph G is the greatest number of colours in a proper colouring of the vertices of G such that for every pair of colours some vertex of the first colour and some vertex of the second colour are adjacent. We prove that almost all graphs Gn with n vertices satisfy n/(k+l) < ψ(Gn) < n/(k–1), where k = k(n) = (log2n)½. We show also that the achromatic number is a ‘global’ invariant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 1 , July 1982 , pp. 21 - 28
Copyright
Copyright © Cambridge Philosophical Society 1982

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

REFERENCES

(1)Bollobás, B. and Erdös, P.Cliques in random graphs. Math. Proc. Cambridge Phil. Soc. 80 (1976), 419–427.CrossRefGoogle Scholar
(2)Bollobás, B.Graph theory - an introductory course. Graduate Texts in Mathematics 63 (Springer-Verlag, New York, Heidelberg and Berlin, 1979).Google Scholar
(3)Bollobas, B., Catlin, P. A. and Erdös, P.Hadwiger's conjecture is true for almost every graph. Europ. J. Combinatorics 1 (1980), 195–199.CrossRefGoogle Scholar
(4)Erdö;s, P. and Spencer, J.Probabilistic methods in combinatorics (Academic Press, New York and London, 1974).Google Scholar
(5)Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J.Correlation inequalities on some partially ordered sets. Communications of Mathematical Physics 22 (1971), 89–103.CrossRefGoogle Scholar
(6)Garey, M. R. and Johnson, D. S.Computers and intractability (Freeman, San Francisco, 1979).Google Scholar
(7)Grimmett, G. R. and McDiarmid, C. J.H. On colouring random graphs. Math. Proc. Cambridge Phil. Soc. 77 (1975), 313–324.CrossRefGoogle Scholar
(8)Harris, T. E.A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Phil. Soc. 56 (1960), 13–20.CrossRefGoogle Scholar
(9)Johnson, D. S. Worst case behaviour of graph coloring algorithms. In Graph Theory and Computing, Proc. Fifth S.E. Conf. on Combinatorics (Utilitatis Mathematica, Winnipeg, 1974), 513–538.Google Scholar
(10)McDiarmid, C. J. H. Determining the chromaticnumber of a graph. Technical Report STAN-CS-76–576, Stanford University, California, 1976.Google Scholar
(11)McDiarmid, C. J. H.Determining the chromatic number of a graph. SIAM J. Computing 8 (1979), 1–14.CrossRefGoogle Scholar
(12)McDiarmid, C. J. H. Colouring random graphs badly. In Graph Theory and Combinatorics (ed. Wilson, R. J.), Pitman Research Notes in Mathematics 34 (1977), pp. 76–86.Google Scholar
(13) McDiarmid, C. J. H. Lower bounds for chromatic numbers of graphs. (To appear.)Google Scholar
(14)Smythe, R. T. and Wierman, J. C. First-passage percolation on the square lattice. Lecture Notes in Mathematics, no. 671 (Springer-Verlag, Berlin, Heidelberg and New York, 1978).Google Scholar