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)=V ₁⋃V ₂ such that G has at most s edges with both incident vertices in V _i . Alon proved that the limsup of $f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)$ tends to infinity as m tends to infinity, establishing a conjecture of Erdős. Bollobás and Scott proposed the following judicious version of Erdős' conjecture: the limsup of $m/4 + \left( {\sqrt {m/32} - g(m)} \right)$ 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 Bollobás-Scott conjecture to k-partitions for odd integers k.