#### Title

Claw-Free Graphs and Separating Independent Sets On 2-Factors

#### Document Type

Article

#### Publication Date

3-2012

#### Publication Title

Journal of Graph Theory

#### DOI

10.1002/jgt.20579

#### ISSN

1097-0118

#### Abstract

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*≥3α^{3}/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.

#### Recommended Citation

Faudree, Ralph J., Colton Magnant, Kenta Ozeki, Kiyoshi Yoshimoto.
2012.
"Claw-Free Graphs and Separating Independent Sets On 2-Factors."
*Journal of Graph Theory*, 69 (3): 251-263.
doi: 10.1002/jgt.20579

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/109