New results on the Zarankiewicz problem

Abstract

Let ( X , Y ) denote a bipartite graph with classes X and Y such that | X | = m and | Y | = n . A complete bipartite subgraph with s vertices in X and t vertices in Y is denoted by K ( s , t ) . The Zarankiewicz problem consists in finding the maximum number of edges, denoted by z ( m , n ; s , t ) , of a bipartite graph ( X , Y ) with | X | = m and | Y | = n without a complete bipartite K ( s , t ) as a subgraph. First, we prove that z ( m , n ; s , t ) = mn - ( m + n - s - t + 1 ) if max { m , n } ⩽ s + t - 1 . Then we characterize the family Z ( m , n ; s , t ) of extremal graphs for the values of parameters described above. Finally, we study the s = t case. We give the exact value of z ( m , n ; t , t ) if 2 t ⩽ n ⩽ 3 t - 1 and we characterize the extremal graphs if either n = 2 t or both 2 t < n ⩽ 3 t - 1 and m ⩽ ⌊ ( 3 t - 1 ) / 2 ⌋ .