Finding dense minors using average degree
Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible t‐vertex minor in graphs of average degree at least t − 1. We show that if G has average degree at least t − 1, it contains a minor on t vertices with at least ( 2 − 1 − o ( 1 ) ) t 2 edges. We show that th...
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| Published in | Journal of graph theory Vol. 108; no. 1; pp. 205 - 223 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Wiley Subscription Services, Inc
01.01.2025
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0364-9024 1097-0118 |
| DOI | 10.1002/jgt.23169 |
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| Abstract | Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible
t‐vertex minor in graphs of average degree at least
t
−
1. We show that if
G has average degree at least
t
−
1, it contains a minor on
t vertices with at least
(
2
−
1
−
o
(
1
)
)
t
2 edges. We show that this cannot be improved beyond
3
4
+
o
(
1
)
t
2. Finally, for
t
≤
6 we exactly determine the number of edges we are guaranteed to find in the densest
t‐vertex minor in graphs of average degree at least
t
−
1. |
|---|---|
| AbstractList | Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible
t‐vertex minor in graphs of average degree at least
t
−
1. We show that if
G has average degree at least
t
−
1, it contains a minor on
t vertices with at least
(
2
−
1
−
o
(
1
)
)
t
2 edges. We show that this cannot be improved beyond
3
4
+
o
(
1
)
t
2. Finally, for
t
≤
6 we exactly determine the number of edges we are guaranteed to find in the densest
t‐vertex minor in graphs of average degree at least
t
−
1. Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible ‐vertex minor in graphs of average degree at least . We show that if has average degree at least , it contains a minor on vertices with at least edges. We show that this cannot be improved beyond . Finally, for we exactly determine the number of edges we are guaranteed to find in the densest ‐vertex minor in graphs of average degree at least . Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible t‐vertex minor in graphs of average degree at least t−1. We show that if G has average degree at least t−1, it contains a minor on t vertices with at least (2−1−o(1))t2 edges. We show that this cannot be improved beyond 34+o(1)t2. Finally, for t≤6 we exactly determine the number of edges we are guaranteed to find in the densest t‐vertex minor in graphs of average degree at least t−1. |
| Author | Turcotte, Jérémie Steiner, Raphael Norin, Sergey Hendrey, Kevin |
| Author_xml | – sequence: 1 givenname: Kevin surname: Hendrey fullname: Hendrey, Kevin organization: Institute for Basic Science (IBS) – sequence: 2 givenname: Sergey surname: Norin fullname: Norin, Sergey organization: McGill University – sequence: 3 givenname: Raphael orcidid: 0000-0002-4234-6136 surname: Steiner fullname: Steiner, Raphael organization: ETH Zürich – sequence: 4 givenname: Jérémie orcidid: 0000-0002-4555-8425 surname: Turcotte fullname: Turcotte, Jérémie email: mail@jeremieturcotte.com organization: McGill University |
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| Notes | Sergey Norin Kevin Hendrey www.math.mcgill.ca/snorin/ Jérémie Turcotte Raphael Steiner www.jeremieturcotte.com sites.google.com/view/raphael-mario-steiner/ sites.google.com/view/kevinhendrey/ ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
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| References | 1927; 10 2010; 65 2023 1968; 178 2022 2020; 94 1943; 88 1986 2022; 37 1964; 153 2017 2016 1967; 174 2014; 75 e_1_2_7_6_1 e_1_2_7_5_1 e_1_2_7_3_1 e_1_2_7_9_1 e_1_2_7_8_1 e_1_2_7_7_1 e_1_2_7_18_1 e_1_2_7_17_1 e_1_2_7_16_1 e_1_2_7_2_1 e_1_2_7_15_1 e_1_2_7_14_1 e_1_2_7_13_1 e_1_2_7_12_1 e_1_2_7_11_1 e_1_2_7_10_1 Hadwiger H. (e_1_2_7_4_1) 1943; 88 |
| References_xml | – year: 1986 – volume: 10 start-page: 96 year: 1927 end-page: 115 article-title: Zur allgemeinen Kurventheorie publication-title: Fund. Math – volume: 37 year: 2022 – volume: 153 start-page: 69 year: 1964 end-page: 80 article-title: Homomorphism theorems for graphs publication-title: Math. Ann – volume: 94 start-page: 206 issue: 2 year: 2020 end-page: 223 article-title: The extremal function for minors publication-title: J. Graph Theory – volume: 88 start-page: 133 year: 1943 end-page: 143 article-title: Über eine Klassifikation der Streckenkomplexe publication-title: Vierteljschr. Naturforsch. Ges. Zärich – year: 2022 – year: 2023 – volume: 174 start-page: 265 year: 1967 end-page: 268 article-title: Homomorphieeigenschaften und mittlere Kantendichte von Graphen publication-title: Math. Ann – start-page: 417 year: 2016 end-page: 437 – volume: 75 start-page: 377 issue: 4 year: 2014 end-page: 386 article-title: Minors in ‐chromatic graphs, II publication-title: J. Graph Theory – year: 2017 – volume: 178 start-page: 154 year: 1968 end-page: 168 article-title: Homomorphiesätze für Graphen publication-title: Math. Ann. – volume: 65 start-page: 343 issue: 4 year: 2010 end-page: 350 article-title: On minors in ‐chromatic graphs publication-title: J. Graph Theory – ident: e_1_2_7_5_1 doi: 10.1002/jgt.20485 – ident: e_1_2_7_6_1 doi: 10.1002/jgt.21744 – volume: 88 start-page: 133 year: 1943 ident: e_1_2_7_4_1 article-title: Über eine Klassifikation der Streckenkomplexe publication-title: Vierteljschr. Naturforsch. Ges. Zärich – ident: e_1_2_7_9_1 doi: 10.4064/fm-10-1-96-115 – ident: e_1_2_7_10_1 – ident: e_1_2_7_16_1 – ident: e_1_2_7_13_1 doi: 10.1002/jgt.22515 – ident: e_1_2_7_2_1 – ident: e_1_2_7_3_1 doi: 10.1007/BF01361708 – ident: e_1_2_7_14_1 – ident: e_1_2_7_7_1 doi: 10.1007/BF01364272 – ident: e_1_2_7_11_1 – ident: e_1_2_7_8_1 doi: 10.1007/BF01350657 – ident: e_1_2_7_18_1 – ident: e_1_2_7_17_1 – ident: e_1_2_7_15_1 doi: 10.1007/978-3-319-32162-2_13 – ident: e_1_2_7_12_1 |
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| Snippet | Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible
t‐vertex minor in graphs of average degree at least
t
−
1. We show... Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible ‐vertex minor in graphs of average degree at least . We show that if... Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible t‐vertex minor in graphs of average degree at least t−1. We show that... |
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| SubjectTerms | Apexes average degree graph minors Graph theory Graphs Hadwiger's conjecture |
| Title | Finding dense minors using average degree |
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