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 $G[N(u)]$ 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 on an induced cycle of length at least $4$ with at most $s$ non-singular edges and with at least $s-5$ locally connected vertices; (ii) for each locally disconnected vertex $v$ of degree $2$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle $C$ with at most $s$ non-singular edges and with at least $s-3$ locally connected vertices and such that $G[V (C)\cap V_{2} (G)]$ is a path or a cycle, then $G$ has a 2-factor, and it is the best possible in some sense. This result generalizes two known results in Faudree, Faudree and Ryjáček (2008) and in Ryjáček, Xiong and Yoshimoto (2010).
The Ondřejov 2-meter Telescope is used either in the Cassegrain or in the coudé focus; the primary focus is not used. The primary goal is the spectroscopy. Nowadays, the Cassegrain focus is equipped with the fiber-fed echelle spectrograph HEROS owned by the Landessternwarte Heidelberg-Konigstuhl. The coudé focus is equipped with the slit spectrograph. Usually, the slit coudé spectrograph is being used in winter due to the lower sensitivity to the weather condition (seeing).
Let $\tau $ be a type of algebras. A valuation of terms of type $\tau $ is a function $v$ assigning to each term $t$ of type $\tau $ a value $v(t) \geq 0$. For $k \geq 1$, an identity $s \approx t$ of type $\tau $ is said to be $k$-normal (with respect to valuation $v$) if either $s = t$ or both $s$ and $t$ have value $\geq k$. Taking $k = 1$ with respect to the usual depth valuation of terms gives the well-known property of normality of identities. A variety is called $k$-normal (with respect to the valuation $v$) if all its identities are $k$-normal. For any variety $V$, there is a least $k$-normal variety $N_k(V)$ containing $V$, namely the variety determined by the set of all $k$-normal identities of $V$. The concept of $k$-normalization was introduced by K. Denecke and S. L. Wismath in their paper (Algebra Univers., 50, 2003, pp.107-128) and an algebraic characterization of the elements of $N_k(V)$ in terms of the algebras in $V$ was given in (Algebra Univers., 51, 2004, pp. 395--409). In this paper we study the algebras of the variety $N_2(V)$ where $V$ is the type $(2,2)$ variety $L$ of lattices and our valuation is the usual depth valuation of terms. We introduce a construction called the {\it $3$-level inflation} of a lattice, and use the order-theoretic properties of lattices to show that the variety $N_2(L)$ is precisely the class of all $3$-level inflations of lattices. We also produce a finite equational basis for the variety $N_2(L)$.