The max-bisection problem is to find a partition of the vertices of agraph into two equal size subsets that maximizes the number of edges withendpoints in both subsets.We obtain new improved approximation ratios for the max-bisection problem onthe low degree k-regular graphs for 3 < k < 8, by deriving some improvedtransformations from a maximum cut into a maximum bisection partition. In thecase of three regular graphs we obtain an approximation ratios of 0.805, and 0.812, respectively.We also present the first polynomial time approximation scheme for themax-bisection problem for planar graphs of a sublinear degree.