Factors of regular graphs

Abstract

A spanning subgraph F of a graph G is called a [ k − 1, k]-factor if k − 1 ≤ d f ( x) ≤ k for all vertices x of G, where d F ( x) denotes the degree of x in F. Tutte proved that if r is an odd integer, then every r-regular graph has a [ k − 1, k]-factor for every integer k, 0 < k < r. We prove that if r is odd and 0 < k ≤ 2r 3 , then every r-regular graph has a [ k − 1, k]-factor each of whose components is regular.