Advances in Applied Mathematics ( IF 1.0 ) Pub Date : 2021-07-19 , DOI: 10.1016/j.aam.2021.102248 George Drummond 1
A matroid is uniform if and only if it has no minor isomorphic to and is paving if and only if it has no minor isomorphic to . This paper considers, more generally, when a matroid M has no -minor for a fixed pair of positive integers . Calling such a matroid -uniform, it is shown that this is equivalent to the condition that every rank- flat of M has nullity less than ℓ. Generalising a result of Rajpal, we prove that for any pair of positive integers and prime power q, only finitely many simple cosimple -representable matroids are -uniform. Consequently, if Rota's Conjecture holds, then for every prime power q, there exists a pair of positive integers such that every excluded minor of -representability is -uniform. We also determine all binary -uniform matroids and show the maximally 3-connected members to be and a particular self-dual matroid . Combined with results of Acketa and Rajpal, this completes the list of binary -uniform matroids for which .
中文翻译:
均匀拟阵的推广
拟阵是一致的当且仅当它没有次要同构于 并且正在铺路当且仅当它没有次要同构到 . 本文更一般地考虑,当拟阵M没有-minor 用于固定的正整数对 . 调用这样的拟阵-uniform,说明这等价于每一个等级的条件-M 的平坦度小于ℓ。概括 Rajpal 的结果,我们证明对于任何对正整数和素数q,只有有限多个简单的余简单-可表示的拟阵是 -制服。因此,如果 Rota's Conjecture 成立,那么对于每个素数q,都存在一对 正整数,使得每个排除的次要 -代表性是 -制服。我们还确定所有二进制-uniform matroids 并显示最多 3 个连接的成员为 和一个特定的自对偶拟阵 . 结合Acketa和Rajpal的结果,这完成了二进制列表-均匀拟阵,其中 .