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 inJournal of graph theory Vol. 108; no. 1; pp. 205 - 223
Main Authors Hendrey, Kevin, Norin, Sergey, Steiner, Raphael, Turcotte, Jérémie
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.01.2025
Subjects
Apexes
average degree
graph minors
Graph theory
Graphs
Hadwiger's conjecture
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ISSN0364-9024
1097-0118
DOI10.1002/jgt.23169

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Summary: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.
Bibliography: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/
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ISSN:0364-9024
1097-0118
DOI:10.1002/jgt.23169