Bipartite b-matching ($F\subseteq E,{\mathrm{deg}}_F(v)\le b_v$)

For a bipartite graph with bipartition $L,R$ and nonnegative integer bounds $b_v$, a bipartite b-matching is an edge subset $F$ whose degree of a vertex is at most $b_v$ at each $v$. Taking $b_v=1$ gives a matching in a graph. To maximize $|F|$, build a flow network with arcs $s\to u$ of flow network edge capacity $b_u$ for $u\in L$, arcs $u\to v$ of flow network edge capacity one for $uv\in E$, and arcs $v\to t$ of flow network edge capacity $b_v$ for $v\in R$. The integral max-flow theorem gives an integral optimum. Select precisely the middle arcs with flow one. Flow conservation identifies their endpoint degrees with the outside arc flows, and the flow value is $|F|$. Conversely, every such edge subset supplies a feasible flow of that value. This proves the reduction and its optimality.