Bipartite graphs with no K6 minor

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 $\epsilon >0$ there are arbitrarily large graphs with average degree at least $8-\epsilon$ 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 $\epsilon >0$ there are arbitrarily large bipartite graphs with average degree at least $8-\epsilon$ 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.