Tree matchings

© 2019 Wiley Periodicals, Inc. An (s, t)-matching in a bipartite graph G = (U, V, E) is a subset of the edges F such that each component of H = (U, V, F) is a tree with at most t edges and each vertex in U has s neighbours in H. We give sharp sufficient neighbourhood-conditions for a bipartite graph to contain an (s, t)-matching. As a special case, we prove a conjecture of Bennett, Bonacina, Galesi, Huynh, Molloy and Wollan.