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 ( k + 1 ) -connected matroids whose symmetric difference between their collections of bases has size at most k , there is a matroid that is obtained from one of these matroids by relaxing n 1 circuit-hyperplanes and from the other by relaxing n 2 circuit-hyperplanes, where n 1 and n 2 are non-negative integers such that n 1 + n 2 ≤ k . In [Matroids with many common bases, Discrete Math. 270 (2003) 193–205], Lemos proved a similar result, where the hypothesis of the matroids being k -connected is replaced by the weaker hypothesis of being vertically k -connected. In this paper, we extend these results.
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