On some graph densities in locally dense graphs

Joonkyung Lee

doi:10.1002/rsa.20974

On some graph densities in locally dense graphs

Joonkyung Lee

Department of Mathematics

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

Abstract

The Kohayakawa–Nagle–Rödl-Schacht conjecture roughly states that every sufficiently large locally d-dense graph G on n vertices must contain at least (1 − o(1))d^|E(H)|n^{|V (H)|} copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.

Original language	English
Pages (from-to)	322-344
Number of pages	23
Journal	Random Structures and Algorithms
Volume	58
Issue number	2
DOIs	https://doi.org/10.1002/rsa.20974
Publication status	Published - 2021 Mar

Bibliographical note

Funding Information:
I would like to thank David Conlon for bringing Conjecture 1.1 to my attention and for many helpful discussions. I am also grateful to Oliver Riordan, Mathias Schacht, and Sasha Sidorenko, who carefully read various versions of this paper and gave useful comments. I would like to thank anonymous referees for careful reports that improved the presentation of the draft, too.

Funding Information:
This research was supported by the European Research Council, ERC Consolidator Grant, PEPCo 724903. Funding information

Publisher Copyright:
© 2020 Wiley Periodicals LLC

All Science Journal Classification (ASJC) codes

Software
Mathematics(all)
Computer Graphics and Computer-Aided Design
Applied Mathematics

Access to Document

10.1002/rsa.20974

Cite this

@article{c60e465be421428e8902b22d94eca960,

title = "On some graph densities in locally dense graphs",

abstract = "The Kohayakawa–Nagle–R{\"o}dl-Schacht conjecture roughly states that every sufficiently large locally d-dense graph G on n vertices must contain at least (1 − o(1))d|E(H)|n|V (H)| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.",

author = "Joonkyung Lee",

note = "Funding Information: I would like to thank David Conlon for bringing Conjecture 1.1 to my attention and for many helpful discussions. I am also grateful to Oliver Riordan, Mathias Schacht, and Sasha Sidorenko, who carefully read various versions of this paper and gave useful comments. I would like to thank anonymous referees for careful reports that improved the presentation of the draft, too. Funding Information: This research was supported by the European Research Council, ERC Consolidator Grant, PEPCo 724903. Funding information Publisher Copyright: {\textcopyright} 2020 Wiley Periodicals LLC",

year = "2021",

month = mar,

doi = "10.1002/rsa.20974",

language = "English",

volume = "58",

pages = "322--344",

journal = "Random Structures and Algorithms",

issn = "1042-9832",

publisher = "John Wiley and Sons Ltd",

number = "2",

}

TY - JOUR

T1 - On some graph densities in locally dense graphs

AU - Lee, Joonkyung

N1 - Funding Information: I would like to thank David Conlon for bringing Conjecture 1.1 to my attention and for many helpful discussions. I am also grateful to Oliver Riordan, Mathias Schacht, and Sasha Sidorenko, who carefully read various versions of this paper and gave useful comments. I would like to thank anonymous referees for careful reports that improved the presentation of the draft, too. Funding Information: This research was supported by the European Research Council, ERC Consolidator Grant, PEPCo 724903. Funding information Publisher Copyright: © 2020 Wiley Periodicals LLC

PY - 2021/3

Y1 - 2021/3

N2 - The Kohayakawa–Nagle–Rödl-Schacht conjecture roughly states that every sufficiently large locally d-dense graph G on n vertices must contain at least (1 − o(1))d|E(H)|n|V (H)| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.

AB - The Kohayakawa–Nagle–Rödl-Schacht conjecture roughly states that every sufficiently large locally d-dense graph G on n vertices must contain at least (1 − o(1))d|E(H)|n|V (H)| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.

UR - http://www.scopus.com/inward/record.url?scp=85097538449&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85097538449&partnerID=8YFLogxK

U2 - 10.1002/rsa.20974

DO - 10.1002/rsa.20974

M3 - Article

AN - SCOPUS:85097538449

SN - 1042-9832

VL - 58

SP - 322

EP - 344

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -

On some graph densities in locally dense graphs

Abstract

Bibliographical note

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this