A (d, d + 1)-graph is a graph whose vertices all have degrees in the set {d, d + 1}. Such a graph is semiregular. An (r, r + 1)-factorization of a graph G is a decomposition of G into (r, r + 1)-factors. For d-regular simple graphs G we say for which x and r G must have an (r, r + 1)-factorization with exactly x (r, r + 1)-factors. We give similar results for (d, d + 1)-simple graphs and for (d...