Bounded transversals in multipartite graphs

Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f -bounded transversals (or f -BT), that is, transversals whose connected components have order at most f . In some sense we search for the the sparsest f -BT-free graphs. We obtain estimates on the smallest maximum degree that 3-partite and 4-partite graphs without 2-BT can have and provide a greatly simplified proof of the best known general lower bound on the smallest maximum degree in f -BT-free graphs.