Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T08:53:58.080Z Has data issue: false hasContentIssue false
  • English
  • Français

Long Monotone Trails in Random Edge-Labellings of Random Graphs

Part of: Graph theory

Published online by Cambridge University Press:  08 October 2019

Omer Angel ,
Asaf Ferber ,
Benny Sudakov  and
Vincent Tassion
Omer Angel*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
Asaf Ferber
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Benny Sudakov
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland
Vincent Tassion
Affiliation:
Department of Mathematics, ETH, 8092 Zurich, Switzerland
*
*Corresponding author. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.

In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).

MSC classification

Primary: 05C38: Paths and cycles
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 22 - 30
Copyright
© Cambridge University Press 2019 

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.)

Footnotes

†

Supported in part by NSERC.

‡

Research is partially supported by an NSF grant 6935855.

§

Research supported in part by SNSF grant 200021-175573.

Research supported in part by the NCCR SwissMAP, funded by the Swiss National Science Foundation.

References

Alon, N. (2003) Problems and results in extremal combinatorics, I. Discrete Math. 273 31–53.CrossRefGoogle Scholar
Bollobás, B. (2001) Random Graphs, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Bucić, M., Kwan, M., Pokrovskiy, A., Sudakov, B., Tran, T. and Wagner, A. (2018) Nearly-linear monotone paths in edge-ordered graphs. Israel J. Math, to appear. arXiv:1809.01468Google Scholar
Calderbank, A. R., Chung, F. R. and Sturtevant, D. G. (1984) Increasing sequences with nonzero block sums and increasing paths in edge-ordered graphs. Discrete Math. 50 15–28.CrossRefGoogle Scholar
Chvátal, V. and Komlós, J. (1971) Some combinatorial theorems on monotonicity. Canad. Math. Bull. 14 151–157.CrossRefGoogle Scholar
Devroye, L. and Reed, B. (1995) On the variance of the height of random binary search trees. SIAM J. Comput. 24 1157–1162.CrossRefGoogle Scholar
Graham, R. L. and Kleitman, D. J. (1973) Increasing paths in edge ordered graphs. Periodica Math. Hungar. 3 141–148.CrossRefGoogle Scholar
Lavrov, M. and Loh, P.-S. (2016) Increasing Hamiltonian paths in random edge orderings. Random Struct. Algorithms. 48 588–611.CrossRefGoogle Scholar
Martinsson, A. (2019) Most edge-orderings of K n have maximal altitude. Random Struct. Algorithms. 54 559–585. arXiv:1605.07204CrossRefGoogle Scholar
McDiarmid, C. (1995) Minimal positions in a branching random walk. Ann. Appl. Probab. 5 128–139.CrossRefGoogle Scholar
Milans, K. G. (2017) Monotone paths in dense edge-ordered graphs. J. Comb. 8 423–437. arXiv:1509.02143Google Scholar
Roditty, Y., Shoham, B. and Yuster, R. (2001) Monotone paths in edge-ordered sparse graphs. Discrete Math. 226 411–417.CrossRefGoogle Scholar
Rödl, V. (1973) Master’s thesis, Charles University.Google Scholar
De Silva, J., Molla, T., Pfender, F., Retter, T. and Tait, M. (2016) Increasing paths in edge-ordered graphs: The hypercube and random graph. Electron. J. Combin. 23 P2.15.CrossRefGoogle Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82 323–339.CrossRefGoogle Scholar
Yuster, R. (2001) Large monotone paths in graphs with bounded degree. Graphs Combin. 17 579–587.CrossRefGoogle Scholar