Liming Xiong

Let $G$ be a simple graph with order $n$ such that every vertex of degree one is adjacent to a vertex of degree at least three. We prove that the line graph $L(G)$ has a 2-factor with at most $\frac{n-1}{3} $ components if every odd branch-bond of $G$ whose member is not a pendent edge has a shortest branch of length two. We also show that $L(G)$ has a 2-factor with at most $\frac{n-3}{4}$ components if we add the condition that $G$ is 2-connected. Finally we show that $L(G)$ has a 2-factor with at most $\frac{n-3}{3}$ components if every odd branch-bond of a 2-connected graph $G$ has a shortest branch of length one. These results are generalities of an implying result of [Choudum and Paulraj, J. Graph Theory 15(1991) 259--265] and [Egawa and Ota, J. Graph Theory 15(1991) 337--344] stating that the line graph of a graph with minimum degree at least three has a 2-factor. These bounds on the number of components in 2-factor are all sharp. We also present a shorter proof of an earlier result.