Matching behaviour is asymptotically normal

Abstract

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.

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Godsil, C.D. Matching behaviour is asymptotically normal. Combinatorica 1, 369–376 (1981). https://doi.org/10.1007/BF02579458

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AMS subject classification (1980)

  • 05 A 15
  • 05 C 50
  • 60 F 05