Let M be a matroid of rank k, B a base, X a set of rank rank(X) < k, and P a perfect matching of the complete bipartite graph (X, B).


Is it true that there exists an edge (x, b) of P augmenting X (that is to say, rank(X + b - x) > rank(X))?


This is true for transversal matroids, and it looks true for graphical matroids but I suspect it to be false in the general case...



Augmention property of matroid along perfect matching
Not always. Take a graph with three edges: $x\_1=e\_2;e\_1$ and a loop $x\_2$. Take $X=\{x\_1,x\_2\}$, $B=\{e\_1,e\_2\}$, perfect matching is $x\_1-e\_1,x\_2-e\_2$.


