Degree sum conditions and a 2-factor with a bounded number of cycles in claw-free graphs
Masaki Kashima
TL;DR
This work advances the understanding of 2-factors with a bounded number of cycles in claw-free graphs by leveraging Ryjáček's closure to reduce the problem to line graphs of triangle-free graphs. The authors prove that for a claw-free graph $G$ of order $n$, if every independent set $I$ satisfies $|I|\le\delta_G(I)-1$ and $\sigma_{k+1}(G)\ge n$, then $G$ contains a $2$-factor with at most $k$ cycles; a notable corollary is that $\delta(G)\ge\alpha(G)+1$ guarantees a $2$-factor with at most $\alpha(G)$ cycles, partially resolving a conjecture of Faudree et al. (2012). The key method combines the closure technique with a line-graph–to–triangle-free graph reduction and dominating-system arguments to control cycle structure. These results enrich the structural theory of claw-free graphs and connect degree-sum conditions with cycle decompositions, offering insight into how local independent-set constraints influence global spanning subgraphs.
Abstract
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, Ryjáček 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 $|I|\leq δ_G(I)-1$ and $G$ satisfies $σ_{k+1}(G)\geq n$, then $G$ has a 2-factor with at most $k$ cycles, where $δ_G(I)$ denotes the minimum degree of the vertices in $I$. As a corollary of the result, we show that every claw-free graph $G$ with $δ(G)\geq α(G)+1$ has a 2-factor with at most $α(G)$ cycles, which partially solves a conjecture by Faudree et al. in 2012.