Matching behaviour is asymptotically normal


Ak-matching in a graphG is a set ofk edges, no two of which have a vertex in common. The number of these inG is writtenp(G, k). Using an idea due to L. H. Harper, we establish a condition under which these numbers are approximately normally distributed. We show that our condition is satisfied ifn=|V(G)| is large compared to the maximum degree Δ of a vertex inG(i.e. Δ=o(n)) orG is a large complete graph. One corollary of these results is that the number of points fixed by a randomly chosen involution in the symmetric groupS is asymptotically normally distributed.

This is a preview of subscription content, log in to check access.


  1. [1]

    E. A. Bender, Central and local limit theorems applied to asymptotic enumeration,J. Combinatorial Theory A,15 (1973), 91–111.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    E. Rodney Canfield, Application of the Berry—Esséen inequality to combinatorial estimates,J. Combinatorial Theory A,28 (1980), 17–25.

    MATH  Article  Google Scholar 

  3. [3]

    W. Feller,An introduction to Probability Theory and its Applications, Vol. 1, Wiley, New York, 1968.

    Google Scholar 

  4. [4]

    W. Feller,An introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1966.

    Google Scholar 

  5. [5]

    C. D. Godsil andI. Gutman, On the theory of the matching polynomial,J. Graph Theory,5 (1981), 137–144.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    C. D. Godsil, Hermite polynomials and a duality relation for matchings polynomials,Combinatorica,1 (1981), 257–262.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    C. D. Godsil, Matchings and walks in graphs,J. Graph Theory, to appear.

  8. [8]

    G. H. Hardy, J. E. Littlewood andG. Pólya,Inequalities, Cambridge University Press, Cambridge (1934).

    Google Scholar 

  9. [9]

    L. H. Harper, Stirling behaviour is asymptotically normal,Ann. Math. Statist. 38 (1967), 410–414.

    MathSciNet  Google Scholar 

  10. [10]

    O. J. Heilmann andE. H. Lieb, Theory of monomer-dimer systems,Comm. Math. Phys. 25 (1972), 190–232.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam (1979).

    Google Scholar 

  12. [12]

    L. Moser andM. Wyman, On solutions ofx d=1 in symmetric groups,Canad. J. Math. 7 (1955), 159–168.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    J. Riordan,An Introduction to Combinatorial Analysis, Wiley, New York (1958).

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Godsil, C.D. Matching behaviour is asymptotically normal. Combinatorica 1, 369–376 (1981).

Download citation

AMS subject classification (1980)

  • 05 A 15
  • 05 C 50
  • 60 F 05