New type degree conditions for a graph to have a 2-factor

A 2-factor of a graph is a 2-regular spanning subgraph. For a graph $G$ and an independent set $I$ of $G$, let $\delta_G(I)$ denote the minimum degree of vertices contained in $I$. We show that (1) if every independent set $I$ of $G$ satisfies $|I|\leq \delta_G(I)-1$, then $G$ has a 2-factor and that (2) if every independent set $I$ of $G$ satisfies $|I|\leq \delta_G(I)$, then $G$ has a 2-factor unless $G$ is isomorphic to a graph in completely determined exceptional graphs. It can be easily shown that the assumption of (1) is a relaxation of the Dirac condition on Hamiltonicity of graphs, and that the assumption of (2) is a relaxation of the Chv\'{a}tal-Erd\H{o}s condition on Hamiltonicity of graphs. Furthermore, for graphs with the assumption of (1), we show some results on a 2-factor with a bounded number of cycles.