Hostname: page-component-76c49bb84f-lvxqv Total loading time: 0 Render date: 2025-07-03T05:23:55.647Z Has data issue: false hasContentIssue false

On the size of temporal cliques in subcritical random temporal graphs

Published online by Cambridge University Press:  26 June 2025

Caelan Atamanchuk
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada
Luc Devroye
Affiliation:
School of Computer Science, McGill University, Montreal, QC, Canada
Gábor Lugosi*
Affiliation:
Department of Economics and Business, Pompeu Fabra University, Barcelona, Spain ICREA, Pg. Lluís Companys 23, Barcelona, 08010, Spain Barcelona School of Economics, Barcelona, Spain
*
Corresponding Author: Gábor Lugosi; Email: gabor.lugosi@gmail.com

Abstract

A random temporal graph is an Erdős-Rényi random graph $G(n,p)$, together with a random ordering of its edges. A path in the graph is called increasing if the edges on the path appear in increasing order. A set $S$ of vertices forms a temporal clique if for all $u,v \in S$, there is an increasing path from $u$ to $v$. Becker, Casteigts, Crescenzi, Kodric, Renken, Raskin and Zamaraev [(2023) Giant components in random temporal graphs. arXiv,2205.14888] proved that if $p=c\log n/n$ for $c\gt 1$, then, with high probability, there is a temporal clique of size $n-o(n)$. On the other hand, for $c\lt 1$, with high probability, the largest temporal clique is of size $o(n)$. In this note, we improve the latter bound by showing that, for $c\lt 1$, the largest temporal clique is of constant size with high probability.

Information

Type
Paper
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Angel, O., Ferber, A., Sudakov, B. and Tassion, V. (2020) Long monotone trails in random edge-labellings of random graphs. Combin. Probab. Comput. 29(1) 2230.CrossRefGoogle Scholar
Becker, R., Casteigts, A., Crescenzi, P., et al. (2023) Giant components in random temporal graphs. arXiv: 2205.14888.Google Scholar
Boyd, D. W. and Steele, J. M. (1979) Random exchanges of information. J. Appl. Probab. 16(3) 657661.CrossRefGoogle Scholar
Broutin, N., Kamčev, N. and Lugosi, G. (2024) Increasing paths in random temporal graphs. arXiv:2306.11401.CrossRefGoogle Scholar
Casteigts, A., Raskin, M., Renken, M. and Zamaraev, V. (2023) Sharp thresholds in random simple temporal graphs. arXiv:2011.03738.Google Scholar
Chvátal, V. and Komlós, J. (1971) Some combinatorial theorems on monotonicity. Can. Math. Bull. 14(2) 151157.CrossRefGoogle Scholar
Dubhashi, D. and Ranjan, D. (1998) Balls and bins: a study in negative dependence. Rand. Struct. Algorithms 13(2) 99124.3.0.CO;2-M>CrossRefGoogle Scholar
Graham, R. L. and Kleitman, D. J. (1973) Increasing paths in edge ordered graphs. Period. Math. Hungar. 3(1-2) 141148.CrossRefGoogle Scholar
Haigh, J. (1981) Random exchanges of information. J. Appl. Probab. 18(3) 743746.CrossRefGoogle Scholar
Lavrov, M. and Loh, P.-S. (2016) Increasing Hamiltonian paths in random edge orderings. Rand. Struct. Algorithms 48(3) 588611.CrossRefGoogle Scholar
Moon, J. W. (1972) Random exchanges of information. Nieuw Arch. Wisk (3) 20 246249.Google Scholar