On the genus of 4-dimensional products of manifolds

We determine bounds for the regular genus of any 4-manifold, which is the product of S^1 by a closed 3-manifold, or a product of two closed surfaces. This is done by an explicit construction of a graph representing the manifold, and by finding a minimal regular imbedding of it. Note that the same construction in dimension 3 enables to find the Heegaard genus of S^1 by a closed surbface, thus improving a result obtained, for the orientable case, by M. Ochiai in Yokohama Math J., 25 (1977), 109-112