Extremal -free bipartite graphs

We approach a well-known topic in extremal graph theory, the so-called Zarankiewicz Problem [K. Zarankiewicz, Problem P 101, Colloq. Math. 2 (1951), 301], which consists in: (i) calculating the maximum number of edges z ( m , n ; s , t ) that a bipartite graph G with partite classes of cardinalities m and n can have such that G is free of a complete bipartite subgraph K ( s , t ) with s vertices in the m class and t vertices in the n class; (ii) describing all the corresponding extremal bipartite graphs having that maximum number of edges. In this paper, the exact value of z ( m , n ; s , t ) is calculated and the corresponding family Z ( m , n ; s , t ) of extremal graphs is characterized when the parameters satisfy certain relationships.