Tree Matchings

An $(s,t)$-matching in a bipartite graph $G=(U,V,E)$ is a subset of the edges $F$ such that each component of $G[F]$ is a tree with at most $t$ edges and each vertex in $U$ has $s$ neighbours in $G[H]$. We give sharp conditions for a bipartite graph to contain an $(s,t)$-matching. As a special case, we prove a conjecture of Bonacina, Galesi, Huynh and Wollan \cite{CNF}.