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.

Mader 定理表明，每个平均度数至少为 8 的图都有一个$K_6$次要的，如果我们用任何更小的常数替换 8，则这是错误的。用最小度替换平均度似乎没有什么区别：我们不知道是否所有最小度至少为 7 的图都具有$K_6$未成年人，但最低程度为六级肯定是不够的。对于每一个$\epsilon >0$存在至少具有平均度的任意大图$8-\epsilon$且最低学位至少为六级，没有$K_6$次要的。

但是如果我们将自己限制在二部图上呢？第一个陈述仍然正确：对于每个$\epsilon >0$存在任意大的二部图，其平均度至少为$8-\epsilon$和不$K_6$次要的。但令人惊讶的是，现在达到最低程度会产生显着的变化。我们将证明每个最小度数至少为 6 的二部图都有一个$K_6$次要的。事实上，二分大部分中的每个顶点的度数至少为 6 就足够了。