{k, r – k}-Factors of r-Regular Graphs

Abstract

For a set $${\mathcal{S}}$$ of positive integers, a spanning subgraph F of a graph G is called an $${\mathcal{S}}$$ -factor of G if $${\deg_F(x) \in \mathcal{S}}$$ for all vertices x of G, where deg F (x) denotes the degree of x in F. We prove the following theorem on {a, b}-factors of regular graphs. Let r ≥ 5 be an odd integer and k be either an even integer such that 2 ≤ k < r/2 or an odd integer such that r/3 ≤ k < r/2. Then every r-regular graph G has a {k, r–k}-factor. Moreover, for every edge e of G, G has a {k, r–k}-factor containing e and another {k, r–k}-factor avoiding e.