(r, R+1)-factorizations of (d, D+1)-graphs

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, d + 1)-pseudographs. We also show that if d ≥ 2r 2 + 3r − 1, then any (d, d + 1)-multigraph (without loops) has an (r, r + 1)-factorization, and we give some information as to the number of (r, r + 1)-factors which can be found in an (r, r + 1)-factorization.