Degree-weighted Hall condition
= Degree-weighted Hall condition
{c}
In a finite bipartite graph, suppose every left vertex has positive degree and each edge joins vertices with $d(x)\geq d(y)$ from left to right. Then $|N(A)|\geq\sum_{x\in A}\sum_{y\sim x}1/d(y)\geq|A|$, so a matching covers the left side.