On judicious bipartitions of graphs

Abstract

For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V1źV2 such that G has at most s edges with both incident vertices in Vi. Alon proved that the limsup of $$f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)$$źb?b2ź4ac2a tends to infinity as m tends to infinity, establishing a conjecture of Erdźs. Bollobas and Scott proposed the following judicious version of Erdźs' conjecture: the limsup of $$m/4 + \left( {\sqrt {m/32} - g(m)} \right)$$m/4+(m/32źg (m)) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobas-Scott conjecture to k-partitions for odd integers k.