Maximizing Several Cuts Simultaneously

Consider two graphs $G_1$ and $G_2$ on the same vertex set $V$ and suppose that $G_i$ has $m_i$ edges. Then there is a bipartition of $V$ into two classes $A$ and $B$ so that, for both $i=1,2$, we have $e_{G_i}(A,B) \geq m_i/2-\sqrt{m_i}$. This gives an approximate answer to a question of Bollobas and Scott. We also prove results about partitions into more than two vertex classes. Our proofs yield polynomial algorithms.