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Bipartite graphs with no K6 minor

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Date

2024-01

Authors

Chudnovsky, Maria
Scott, Alex
Seymour, Paul
Spirkl, Sophie

Advisor

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Publisher

Elsevier

Abstract

A theorem of Mader shows that every graph with average degree at least eight has a K6 minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have K6 minors, but minimum degree six is certainly not enough. For every ε > 0 there are arbitrarily large graphs with average degree at least 8 − ε and minimum degree at least six, with no K6 minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every ε > 0 there are arbitrarily large bipartite graphs with average degree at least 8 − ε and no K6 minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a K6 minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

Description

This publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2023.08.005 © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC-BY license (http://creativecommons.org/licenses/by/4.0/).

Keywords

minors, bipartite, edge-density

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