Matching preclusion and conditional matching preclusion for bipartite interconnection networks II: Cayley graphs generated by transposition trees and hyper‐stars

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. It is natural to look for obstruction sets beyond those induced by a single vertex. The conditional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph with no isolated vertices that has no perfect matchings. In this companion paper of Cheng et al. (Networks (NET 1554)), we find these numbers for a number of popular interconnection networks including hypercubes, star graphs, Cayley graphs generated by transposition trees and hyper‐stars. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/91319/1/20441_ftp.pd

Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees

AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults

Fault diagnosability of regular graphs

An interconnection network\u27s diagnosability is an important measure of its self-diagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the $h$-good-neighbor conditional diagnosability, which requires that every fault-free node has at least $h$ fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The {\it $h$-good-neighbor diagnosability} under the PMC (resp. MM*) model of a graph $G$, denoted by $t_h^{PMC}(G)$ (resp. $t_h^{MM^*}(G)$), is the maximum value of $t$ such that $G$ is $h$-good-neighbor $t$-diagnosable under the PMC (resp. MM*) model. In this paper, we study the $2$-good-neighbor diagnosability of some general $k$-regular $k$-connected graphs $G$ under the PMC model and the MM* model. The main result $t_2^{PMC}(G)=t_2^{MM^*}(G)=g(k-1)-1$ with some acceptable conditions is obtained, where $g$ is the girth of $G$. Furthermore, the following new results under the two models are obtained: $t_2^{PMC}(HS_n)=t_2^{MM^*}(HS_n)=4n-5$ for the hierarchical star network $HS_n$, $t_2^{PMC}(S_n^2)=t_2^{MM^*}(S_n^2)=6n-13$ for the split-star networks $S_n^2$ and $t_2^{PMC}(\Gamma_{n}(\Delta))=t_2^{MM^*}(\Gamma_{n}(\Delta))=6n-16$ for the Cayley graph generated by the $2$-tree $\Gamma_{n}(\Delta)$