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An edge of $G$ is singular if it does not lie on any triangle of $G$; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G\left[N\right(u\left)\right]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies...

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph. Gould and Hynds (1999) showed a well-known characterization of $G$ by its line graph $L\left(G\right)$ that has a 2-factor. In this paper, by defining two operations, we present a characterization for a graph $G$ to have a 2-factor in its line graph $L\left(G\right).$ A graph $G$ is called $N^2$-locally connected if for every vertex $x\in V\left(G\right),$ $G\left[\{y\in V\left(G\right)1\le {\mathrm{dist}}_G(x,y)\le 2\}\right]$ is connected. By applying the new characterization, we prove that every claw-free graph in which every edge lies on a cycle of length at most five and in which every vertex...