A matroid view of key theorems for edge-swapping algorithms

We demonstrate that two key theorems of Amaldi et al. (Math Methods Oper Res 69:205-223, 2009), which they presented with rather complicated proofs, can be more easily and cleanly established using a simple and classical property of binary matroids. Besides a simpler proof, we see that both of these key results are manifestations of the same essential property.