The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs

@article{McKay1990TheAN,
  title={The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs},
  author={Brendan D. McKay},
  journal={Combinatorica},
  year={1990},
  volume={10},
  pages={367-377}
}
  • B. McKay
  • Published 1 December 1990
  • Mathematics, Computer Science
  • Combinatorica
LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn→∞,, for anyε>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals. 

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References

SHOWING 1-6 OF 6 REFERENCES
APPLICATIONS OF A TECHNIQUE FOR LABELLED ENUMERATION
A technique involving summation over roots of unity was used by Liskovec in 1971 to count labelled regular tournaments. The same method is used here to count regular tournaments to 21 vertices,
Asymptotic Enumeration by Degree Sequence of Graphs of High Degree
TLDR
This work considers the estimation of the number of labelled simple graphs with degree sequence d 1, d 2, . . . , d n by using an n-dimensional Cauchy integral and gives as a corollary the asymptotic joint distribution function of the degrees of a random graph.
Random regular tournaments, Period
  • Math. Hungar
  • 1974
Liskovec, The number of eulerian digraphs and regular tournaments (Russian
  • Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk,
  • 1971
The number of eulerian digraphs and regular tournaments (Russian) Vesci Akad
  • Navuk BSSR Ser. Fiz.-Mat. Navuk
  • 1971