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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≥2*n*/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.

*Journal of Graph Theory*Vol. 69 Iss. 3 (2012) p. 251 - 263 ISSN: 1097-0118

Available at: http://works.bepress.com/colton_magnant/31/