On the number of common bases of two matroids
Let \(M_ i=(E,F_ i)\), \(i=1,2\), be two simple matroids on the same finite set E \((F_ i\) being the family of independent sets for \(M_ i)\). By studying the dimension of the intersection of the polytope \(K(M_ i)\) representing \(M_ i\), \(i=1,2\), (that is, \(K(M_ i)\) is the polytope having as extreme points the representative vectors of the bases of \(M_ i)\), the authors obtain \(| E| -(d_ 1+d_ 2)+2\) for the least number of common bases of \(M_ 1\) and \(M_ 2\), assuming there exists at least one basis in common, where \(d_ i\) is the index of decomposition of \(M_ i\). A number of interesting applications to graph theory are given.
scientific article; zbMATH DE number 3859142