Claw-free graphs and 2-factors that separate independent vertices

In this article, we prove that a line graph with minimum degree δ ≥ 7 has a spanning subgraph in which every component is a clique of order at least three. This implies that if G is a line graph with δ ≥ 7, then for any independent set S there is a 2-factor of G such that each cycle contains at most one vertex of S. This supports the conjecture that δ ≥ 5 is sufficient to imply the existence of such a 2-factor in the larger class of claw-free graphs. It is also shown that if G is a claw-free graph of order n and independence number α with δ ≥ 2n/α−2 and n ≥ 3α3/2, then for any maximum independent set S, G has a 2-factor with α cycles such that each cycle contains one vertex of S. This is in support of a conjecture that δ ≥ n/α ≥ 5 is sufficient to imply the existence of a 2-factor with α cycles, each containing one vertex of a maximum independent set.