Bipartite graphs with no K₆ minor

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 - e 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.(c) 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/).