Tree matchings
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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].
[1] P. Hall. On Representatives of Subsets , 1935 .