论文标题
二分图，没有$ k_6 $ binor
Bipartite graphs with no $K_6$ minor
论文作者
论文摘要
Mader的定理表明，平均度至少八个的每个图都有$ k_6 $的未成年人，如果我们替换八个较小的常数，这是错误的。替换最低学位的平均度似乎没有什么不同：我们不知道所有最低学位的图表是否至少七个$ k_6 $ nminors，但最低学位肯定还不够。对于每$ c> 0 $，都有任意的图形，平均程度至少为$ 8-C $，至少六个至少六个，没有$ k_6 $ binor。 但是，如果我们将自己限制在二分图上怎么办？第一个语句仍然是正确的：每$ c> 0 $都有任意大型的两分图，平均程度至少为$ 8-c $，而没有$ k_6 $ binor。但令人惊讶的是，现在至最低学位有很大的不同。我们将证明，每个具有最低学位至少六个的两分图都有$ k_6 $的未成年人。确实，足以使两者的大部分中的每个顶点至少具有至少六个。
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.
