Abstract
In this paper, we apply the Turán sieve and the simple sieve developed by R. Murty and the first author to study problems in random graph theory. In particular, we obtain upper and lower bounds on the probability of a graph on n vertices having diameter 2 (or diameter 3 in the case of bipartite graphs) with edge probability p where the edges are chosen independently. An interesting feature revealed in these results is that the Turán sieve and the simple sieve “almost completely” complement each other. As a corollary to our result, we note that the probability of a random graph having diameter 2 approaches 1 as \(n\rightarrow \infty \) for constant edge probability \(p=1/2\).
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References
Bollobás, Béla.: The diameter of random graphs. Trans. Am. Math. Soc. 267(1), 41–52 (1981)
Chow, Timothy Y.: The combinatorics behind number-theoretic sieves. Adv. Math. 138(2), 293–305 (1998)
Hardy, G.H., Ramanujan, S.: The normal number of prime factors of a number \(n\), The Quarterly Journal of. Pure Appl. Math. 48, 76–97 (1917)
Gilbert, Edgar N.: Random graphs. Ann. Math. Stat. 30(4), 1141–1144 (1959)
Kuo, Wentang, Liu, Yu.-Ru., Ribas, Sávio., Zhou, Kevin: The shifted Turán sieve method on tournaments. Can. Math. Bull. 62(4), 841–855 (2019)
Kuo, Wentang, Liu, Yu.-Ru., Ribas, Sávio., Zhou, Kevin: The shifted Turán sieve method on tournaments. Discrete Math. 344(12), 112602 (2021)
Liu, Yu.-Ru.: and M. Ram Murty, Sieve methods in combinatorics, J. Combinatorial Theory Ser. A 111(1), 1–23 (2005)
Liu, Yu-Ru., Saunders, J.C.: Sieve Methods in Random Graph Theory, arXiv preprint arXiv:1805.11153 (2018)
Turán, Pál.: On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9, 274–276 (1934)
Wilson, R.J.: The Selberg sieve for a lattice, Combinatorial Theory and its Applications, 1141–1149 (1970)
Acknowledgements
The results in the paper are part of the Ph.D. thesis of the second author. He would like to thank his co-supervisor, Kevin Hare, and his thesis committee members, Karl Dilcher, David McKinnon, Jeffrey Shallit, and Cam Stewart for their helpful suggestions about this project. He would also like to thank the University of Waterloo for the awards of two Queen Elizabeth II Graduate Scholarship in Science and Technology and the Azrieli Foundation for the award of an Azrieli International Postdoctoral Fellowship, as well as the University of Calgary also for the award of a Postdoctoral Fellowship. The authors are grateful for many valuable comments from the referee, particularly the arguments to shorten the proof of Corollary 2.
Funding
This research was supported by the Department of Pure Mathematics at the University of Waterloo, two Graduate Scholarships in Science and Technology, an Azrieli International Postdoctoral Fellowship, and the Department of Mathematics and Statistics at the University of Calgary for the award of a Postdoctoral Fellowship.
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Queen Elizabeth II Graduate Scholarship in Science and Technology, Azrieli International Postdoctoral Fellowship.
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Liu, YR., Saunders, J.C. Sieve Methods in Random Graph Theory. Graphs and Combinatorics 39, 39 (2023). https://doi.org/10.1007/s00373-023-02635-x
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DOI: https://doi.org/10.1007/s00373-023-02635-x