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 [1].