The upper bound of the number of cycles in a 2‐factor of a line graph

Let G be a simple graph with order n and minimum degree at least two. In this paper, we prove that if every odd branch‐bond in G has an edge‐branch, then its line graph has a 2‐factor with at most ${{3n - 2}\over {8}}$ components. For a simple graph with minimum degree at least three also, the same conclusion holds. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 72–82, 2007