Non-separating 2-factors of an even-regular graph

For a 2-factor F of a connected graph G, we consider G − F , which is the graph obtained from G by removing all the edges of F . If G − F is connected, F is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r -regular connected graph G to have such a 2-factor. As a result, we show that a 2r -regular connected graph G has a non-separating 2-factor whenever the number of vertices of G does not exceed 2r2 + r . c © 2007 Elsevier B.V. All rights reserved.