Bipartite graphs with no $K_6$ minor

TL;DR

This paper investigates the conditions under which bipartite graphs contain a $K_6$ minor, revealing that minimum degree constraints in bipartite graphs are sufficient for the existence of such minors, unlike in general graphs.

Contribution

It proves that bipartite graphs with minimum degree at least six always contain a $K_6$ minor, highlighting a key difference from non-bipartite graphs.

Findings

Bipartite graphs with minimum degree ≥6 always contain a $K_6$ minor.

Large bipartite graphs with average degree close to 8 can lack a $K_6$ minor.

Minimum degree constraints are more significant in bipartite graphs for $K_6$ minors.

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.