Degree sum conditions and a 2-factor with a bounded number of cycles in claw-free graphs

A claw-free graph is a graph that does not contain $K_{1,3}$ as an induced subgraph, and a 2-factor is a 2-regular spanning subgraph of a graph. In 1997, Ryj\'{a}\v{c}ek introduced the closure concept of claw-free graphs, and Hamilton cycles and related structures in claw-free graphs have been intensively studied via the closure concept. In this paper, using the closure concept, we show that for a claw-free graph $G$ of order $n$, if every independent set $I$ of $G$ satisfies $|I|\leq \delta_G(I)-1$ and $G$ satisfies $\sigma_{k+1}(G)\geq n$, then $G$ has a 2-factor with at most $k$ cycles, where $\delta_G(I)$ denotes the minimum degree of the vertices in $I$. As a corollary of the result, we show that every claw-free graph $G$ with $\delta(G)\geq \alpha(G)+1$ has a 2-factor with at most $\alpha(G)$ cycles, which partially solves a conjecture by Faudree et al. in 2012.