 
Summary: Bipartite Subgraphs of Integer Weighted Graphs
Noga Alon
Eran Halperin
February 22, 2002
Abstract
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) = n2
4 + min( n
2 , 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 m2
4 n
2 .
1 Introduction
All graphs in this paper contain no loops. For a simple graph G = (V, E), let f(G) denote the
maximum number of edges in a bipartite subgraph of G. For every p > 0, let g(p) denote the
minimum value of f(G), as G ranges over all simple graphs with p edges. Thus, g(p) is the largest
