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).
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 $N_2(L)$.
Římské vojenské tažení proti Marobudovi v r. 6 po Kr. představuje nejstarší přesně datovanou historickou událost vztahující se k České kotlině. Při příležitosti dvoutisíciletého výročí této události se autor zamýšlí nad metodickými problémy bádání o starší době římské v Čechách. Na příkladech chronologie archeologických horizontů, migrace etnických jednotek, římských importů a právě římského tažení v r. 6 je v článku poukázáno na fakt, že česká archeologie tradičně upřednostňuje písemné prameny před archeologickými. Mnohé údaje, o které se badatelé opírají, však nejsou v písemných pramenech doložitelné. Často se jedná pouze o domněnky a interpretace historiků. Článek upozorňuje na metodologickou neúnosnost vytváření archeologických konstrukcí, které jsou závislé na takovýchto domněnkách. and The Roman military campaign against Maroboduus in the year 6 AD is the earliest accurately dated historical event linked to the Bohemian Basin. On the occasion of its 2000th anniversary, the author considers the methodological problems attendant upon research into the early Roman period in Bohemia. Taking examples from the chronology of archaeological horizons, the migrations of ethnic units, Roman imports and the Roman campaign of 6 AD itself, the article demonstrates the fact that Czech archaeology has traditionally prioritised written over archaeological sources. Much if the data on which researchers rely, however, cannot be proven in the written record: often, they are merely the conjectures and interpretations of historians. This article highlights the methodological unjustifiability of creating archaeological constructs that are dependent on such conjectures.