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

Super edge-connectivity and matching preclusion of data center networks

Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$\lambda$ for $k\geq2$ and $n\geq2$, super-$\lambda_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$\lambda_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.Comment: 20 pages, 1 figur

The Nature Diagnosability of Bubble-sort Star Graphs under the PMC Model and MM Model

Many multiprocessor systems have interconnection networks as underlying topologies and an interconnection network is usually represented by a graph where nodes represent processors and links represent communication links between processors. No fault set can contain all the neighbors of any fault-free vertex in the system, which is called the nature diagnosability of the system. Diagnosability of a multiprocessor system is one important study topic. As a famous topology structure of interconnection networks, the -dimensionalnbsp bubble-sort star graph nbsphas many good properties. In this paper, we prove that the nature diagnosability of nbspis nbspunder the PMC model for , the nature diagnosability of nbspis nbspunder the MM model for