Nearly-linear monotone paths in edge-ordered graphs

Matija Bucić; Matthew Kwan; Alexey Pokrovskiy; Benny Sudakov; Tuan Tran; Adam Zsolt Wagner

doi:10.1007/s11856-020-2035-7

Nearly-linear monotone paths in edge-ordered graphs

Matija Bucić, Matthew Kwan, Alexey Pokrovskiy, Benny Sudakov, Tuan Tran, Adam Zsolt Wagner

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

How long a monotone path can one always find in any edge-ordering of the complete graph K_n? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1-o(1).

Original language	English (US)
Pages (from-to)	663-685
Number of pages	23
Journal	Israel Journal of Mathematics
Volume	238
Issue number	2
DOIs	https://doi.org/10.1007/s11856-020-2035-7
State	Published - Jul 1 2020
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s11856-020-2035-7

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title = "Nearly-linear monotone paths in edge-ordered graphs",

abstract = "How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chv{\'a}tal and Koml{\'o}s in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).",

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AU - Bucić, Matija

AU - Kwan, Matthew

AU - Pokrovskiy, Alexey

AU - Sudakov, Benny

AU - Tran, Tuan

AU - Wagner, Adam Zsolt

PY - 2020/7/1

Y1 - 2020/7/1

N2 - How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).

AB - How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvátal and Komlós in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n2/3-o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n1-o(1).

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JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

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Nearly-linear monotone paths in edge-ordered graphs

Abstract

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