On the number of (r,r+1)- factors in an (r,r+1)-factorization of a simple graph

For integers d≥0, s≥0, a (d, d+s)‐graph is a graph in which the degrees of all the vertices lie in the set {d, d+1, …, d+s}. For an integer r≥0, an (r, r+1)‐factor of a graph G is a spanning (r, r+1)‐subgraph of G. An (r, r+1)‐factorization of a graph G is the expression of G as the edge‐disjoint union of (r, r+1)‐factors. For integers r, s≥0, t≥1, let f(r, s, t) be the smallest integer such that, for each integer d≥f(r, s, t), each simple (d, d+s) ‐graph has an (r, r+1) ‐factorization with x (r, r+1) ‐factors for at least t different values of x. In this note we evaluate f(r, s, t). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 257‐268, 2009