Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

The Bubble-sort graph $BS_n,\,n\geqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set $\{(1 2), (2 3),\ldots, (n-1 n)\}$. It is a bipartite graph containing all even cycles of length $\ell$, where $4\leqslant \ell\leqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,\,n\geqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.Comment: 13 pages, 7 figure

The LS category of the product of lens spaces

We reduced Rudyak's conjecture that a degree one map between closed manifolds cannot raise the Lusternik-Schnirelmann category to the computation of the category of the product of two lens spaces $L^n_p\times L_q^n$ with relatively prime $p$ and $q$. We have computed $cat(L^n_p\times L^n_q)$ for values of $p,q>n/2$. It turns out that our computation supports the conjecture. For spin manifolds $M$ we establish a criterion for the equality $cat M=dim M-1$ which is a K-theoretic refinement of the Katz-Rudyak criterion for $cat M=dim M$. We apply it to obtain the inequality $cat(L^n_p\times L^n_q)\le 2n-2$ for all $n$ and odd relatively prime $p$ and $q$

Leavitt path algebras: Graded direct-finiteness and graded Σ\Sigma-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded $\Sigma$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $\Sigma$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $\Sigma$-$V$ ring $L$% . When the graph $E$ is finite, we show that $L$ is a graded $\Sigma$-$V$ ring $\Longleftrightarrow L$ is graded directly-finite $\Longleftrightarrow L$ has bounded index of nilpotence $\Longleftrightarrow$ $L$ is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph $E$ is infinite. Following this, we also characterize Leavitt path algebras $L$ which are non-graded $\Sigma$-$V$ rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra $L$ is a graded $\Sigma$-$V$ ring, then $L$ is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Va\v{s} \cite{V} on directly-finite Leavitt path algebras.Comment: 21 page

On Linkedness of Cartesian Product of Graphs

We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of product of paths and product of cycles