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

For a 2-factor FF of a connected graph GG, we consider G−FG−F, which is the graph obtained from GG by removing all the edges of FF. If G−FG−F is connected, FF is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r2r-regular connected graph GG to have such a 2-factor. As a result, we show that a 2r2r-regular connected graph GG has a non-separating 2-factor whenever the number of vertices of GG does not exceed 2r2+r2r2+r.