Perfect Matching Lemma (Matroid Exchange Graph)

combinatorics computation

Definition

Perfect Matching Lemma (Matroid Exchange Graph)

Let $\mathcal{M}=(E,\mathcal{I})$ be a matroid. For two independent sets $S,T\in \mathcal{I}$ with $|S|=|T|$, the exchange graph $G_{\mathcal{M}}(S)$ has a perfect matching $S\setminus T$ and $T\setminus S$.