Abstract
The exact number Ōp of self-converse oriented graphs on p points has been known for some time; we give the first asymptotic analysis of Ōp. The formula for Ōp was found by an application of Burnside's Lemma, and consists of a sum of terms corresponding to the partitions of p. The usual pattern in such cases is for one term to dominate the others when p is large. For Ōp there is no single term which is dominant. Instead an infinite family of terms must all be considered in the asymptotic treatment. This leads to a much more complicated asymptotic form for Ōp than the ones obtaining for graphs, digraphs, self-complementary graphs, self-complementary digraphs, oriented graphs, tournaments, or self-complementary tournaments. The values of Ōp are tabulated for p up to 27. A sample is then compared with the corresponding sums of dominant terms and final asymptotic expressions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Edward A. Bender, "Asymptotic methods in enumeration", SIAM Review 16 (1974), 485–515.
Frank Harary and Edgar M. Palmer, Graphical Enumeration. Academic Press, N.Y., 1973.
Frank Harary, Edgar M. Palmer, Robert W. Robinson, and Allen J. Schwenk, "Enumeration of graphs with signed points and lines", J. Graph Theory, to appear.
Ronald Fagin, "The number of finite relational structures", Discrete Math, to appear.
John Moon, Topics on Tournaments. Holt, N.Y., 1968.
Edgar M. Palmer, "Asymptotic formulas for the number of self-complementary graphs and digraphs", Mathematika 17 (1970), 85–90.
W. Oberschelp, "Structurzahlen in endlichen relationssystemen", Contributions to Mathematical Logic (Proceedings of Logic Colloquium), Hanover, 1966, 199–213.
W. Oberschelp, "Kombinatorische Anzahlbestimmungen in Relationen", Math. Ann. 174 (1967), 53–58.
M.R. Sridharan, "Self-complementary and self-converse oriented graphs", Indag. Math. 32 (1970), 441–447.
Detlef Wille, "Asymptotic formulas for the number of oriented graphs", J. Combinatorial Theory (B) 21 (1976), 270–274.
Edward M. Wright, "Counting coloured graphs", Canad. J. Math. 13 (1961), 683–693.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1978 Springer-Verlag
About this paper
Cite this paper
Robinson, R.W. (1978). Asymptotic number of self-converse oriented graphs. In: Holton, D.A., Seberry, J. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062540
Download citation
DOI: https://doi.org/10.1007/BFb0062540
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-08953-7
Online ISBN: 978-3-540-35702-5
eBook Packages: Springer Book Archive