Restricted t-matchings in bipartite graphs

Given a simple bipartite graph G and an integer t ≥ 2, we derive a formula for the maximum number of edges in a subgraph H of G so that H contains no node of degree larger than t and H contains no complete bipartite graph Kt,t as a subgraph. In the special case t = 2 this fomula was proved earlier by Z. Király [6], sharpening a result of D. Hartvigsen [4]. For any integer t ≥ 2, we also determine the maximum number of edges in a subgraph of G that contains no complete bipartite graph, as a subgraph, with more than t nodes. The proofs are based on a general min-max result of [2] concerning crossing bi-supermodular functions.