Bipartite subgraphs of integer weighted graphs

For every integer p > 0 let f(p) be the minimum possible value of the maximum weight of a cut in an integer weighted graph with total weight p. It is shown that for every large n and every m < n, f( ( n 2 ) + m) = b 2 4 c + min(d n 2 e, f(m)). This supplies the precise value of f(p) for many values of p including, e.g, all p = ( n 2 ) + ( m 2 ) when n is large enough and m 2 4 ≤ n 2 .