Affiliation: | School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, England, United Kingdom Department of Mathematics, University of Reading, Whiteknights, Reading, RG6 6AX, England, United Kingdom |
Abstract: | For d≥1, s≥0 a (d,d+s)-graph is a graph whose degrees all lie in the interval {d,d+1,…,d+s}. For r≥1, a≥0 an (r,r+a)-factor of a graph G is a spanning (r,r+a)-subgraph of G. An (r,r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r,r+a)-factors. A graph is (r,r+a)-factorable if it has an (r,r+a)-factorization.We prove a number of results about (r,r+a)-factorizations of (d,d+s)-bipartite multigraphs and of (d,d+s)-pseudographs (multigraphs with loops permitted). For example, for t≥1 let β(r,s,a,t) be the least integer such that, if d≥β(r,s,a,t) then every (d,d+s)-bipartite multigraph G is (r,r+a)-factorable with x(r,r+a)-factors for at least t different values of x. Then we show that |