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arXiv · 2204.10119

Bipartite graphs with no $K_6$ minor

Maria Chudnovsky·Alex Scott·Paul Seymour·Sophie Spirkl
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Abstract

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ 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 $K_6$ minors, but minimum degree six is certainly not enough. For every $c>0$ there are arbitrarily large graphs with average degree at least $8-c$ and minimum degree at least six, with no $K_6$ minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $c>0$ there are arbitrarily large bipartite graphs with average degree at least $8-c$ and no $K_6$ 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 $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

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Maria Chudnovsky, Alex Scott, Paul Seymour, Sophie Spirkl. 2022-04-21. Bipartite graphs with no $K_6$ minor. https://arxiv.org/abs/2204.10119

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