Cutting two graphs simultaneously

Consider two graphs, G1 and G2, on the same vertex set V , with |V | = n and Gi having mi edges for i = 1, 2. We give a simple algorithm that partitions V into sets A and B such that eG1(A,B) ≥ m1/2 and eG2(A,B) ≥ m2/2−∆(G2)/2. We also show, using a probabilistic method, that if G1 and G2 belong to certain classes of graphs, (for instance, if G1 and G2 both have a density of at least 2/3, or if G1 and G2 are both regular of degree at most (n/16) − 6 with n sufficiently large) then we can find a partition of V into sets A and B such that eGi(A,B) ≥ mi/2 for i = 1, 2.