Matroids with few non-common bases

Abstract: In [On Mills''s conjecture on matroids with many common bases, Discrete Math. 240 (2001) 271–276], Lemos proved a conjecture of Mills [On matroids with many common bases, Discrete Math. 203 (1999) 195–205]: for two -connected matroids whose symmetric difference between their collections of bases has size at most , there is a matroid that is obtained from one of these matroids by relaxing circuit-hyperplanes and from the other by relaxing circuit-hyperplanes, where and are non-negative integers such that . In [Matroids with many common bases, Discrete Math. 270 (2003) 193–205], Lemos proved a similar result, where the hypothesis of the matroids being -connected is replaced by the weaker hypothesis of being vertically -connected. In this paper, we extend these results. [Copyright &y& Elsevier]