Conditional connectivity of Cayley graphs generated by transposition trees

Given a graph G and a non-negative integer h, the Rh-(edge)connectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has minimum degree at least h. Similarly, given a non-negative integer g, the g-(edge)extraconnectivity of G is the minimum cardinality of a set of (edges)vertices of G, if any, whose deletion disconnects G, and every remaining component has more than g vertices. In this paper, we determine R₂-(edge)connectivity and 2-extra(edge)connectivity of Cayley graphs generated by transposition trees.     

A probabilistic algorithm for vertex connectivity of graphs

A probabilistic algorithm is presented which computes the vertex connectivity of an undirected graph G = (V,E) in expected time $\text{O((-log ε|V|}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{|E|)}$ with error probability at most e provided that |E|<frcase|1/2d|V|² for some universal constant d<1.     

Edge-fault-tolerant bipancyclicity of Cayley graphs generated by transposition-generating trees

The Cayley graphs on the symmetric group plays an important role in the study of Cayley graphs as interconnection networks. Let Cay(S_n, B) be the Cayley graphs generated by transposition-generating trees. It is known that for any F?E(Cay(S_n, B)), if |F|≤n?3 and n≥4, then there exists a hamiltonian cycle in Cay(Sn, B)?F. In this paper, we show that Cay(S_n, B)?F is bipancyclic if Cay(S_n, B) is not a star graph, for n≥4 and |F|≤n?3.     

Linearly many faults in Cayley graphs generated by transposition trees

We prove that when linearly many vertices are deleted in a Cayley graph generated by a transposition tree, the resulting graph has a large connected component containing almost all remaining vertices.     

Honeycomb toroidal graphs are Cayley graphs

There is a particular family of trivalent vertex-transitive graphs that have been called generalized honeycomb tori by some and brick products by others. They have been studied as hexagonal embeddings on the torus as well. We show that all these graphs are Cayley graphs on generalized dihedral groups.     

泡形互连网络的条件连通性度量

n维泡形网络是设计大规模多处理机系统时最常用的互连网络拓扑结构之一,它以n维泡形图Bn为数学模型。F是连通图G的顶点子集,使得G-F不再连通且G-F的每个连通分支都有至少有n个顶点的F的势叫做G的Rk连通度。Rk连通度是衡量网络可靠性的一个重要参数。一般来说,网络的Rk连通度越大,其可靠性越高。研究了n维泡形网络的 k连通性;证明了在n维泡形网络中,当n≥3时,其R1连通度为2n-4;当n≥4 时,其R2连通度为4n-12。     

Component connectivity of generalized Petersen graphs

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λ_r (G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this paper we establish bounds of the r component edge connectivity of an important family of interconnection network models, the generalized Petersen graphs GP(n, k) in which n and k are relatively prime integers.     

Nowhere-zero 3-flows in dihedral Cayley graphs

In this paper we prove that every k-valent Cayley graph of a dihedral group, where k?4, admits a nowhere-zero 3-flow.     

A kind of conditional fault tolerance of alternating group graphs

A vertex subset F is a k-restricted vertex-cut of a connected graph G if G−F is disconnected and every vertex in G−F has at least k good neighbors in G−F. The cardinality of the minimum k-restricted vertex-cut of G is the k-restricted connectivity of G, denoted by κk(G). This parameter measures a kind of conditional fault tolerance of networks. In this paper, we show that for the n-dimensional alternating group graph AGn, κ²(AG₄)=4 and κ²(AGn)=6n−18 for n?5.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

Super p-restricted edge connectivity of line graphs

Super restricted connectivity and super restricted edge connectivity are more refined network reliability indices than connectivity and edge connectivity. In this paper, we first introduce the concepts of super restricted connectivity and super restricted edge connectivity and then give a property of graphs with equal p-restricted edge connectivity and 1-p-restricted connectivity. Applying this property, we show a sufficient condition for a line graph to be super p-restricted edge-connected and a relationship between super 1-p-restricted connected graphs and super p-restricted edge-connected graphs.     

Edge-fault-tolerant Hamiltonicity of pancake graphs under the conditional fault model

The conditional fault model imposes a constraint on the fault distribution. For example, the most commonly imposed constraint for edge faults is that each vertex is incident with two or more non-faulty edges. In this paper, subject to this constraint, we show that an n$n$-dimensional pancake graph can tolerate up to 2n−7$2n-7$ edge faults, while retaining a fault-free Hamiltonian cycle, where n≥4$n\ge 4$. Previously, at most n−3$n-3$ edge faults can be tolerated for the same problem, if the edge faults may occur anywhere without imposing any constraint.     

A channel assignment problem for optical networks modelled by Cayley graphs

A problem arising from a recent study of scalability of optical networks seeks to assign channels to the vertices of a network so that vertices distance 2 apart receive distinct channels. In this paper we introduce a general channel assignment scheme for Cayley graphs on abelian groups, and derive upper bounds for the minimum number of channels needed for such graphs. As application we give a systematic way of producing near-optimal channel assignments for connected graphs admitting a vertex-transitive abelian group of automorphisms. Hypercubes are examples of such graphs, and for them our near-optimal upper bound gives rise to the one obtained recently by Wan.     

Embedding Hamiltonian cycles in alternating group graphs under conditional fault model

In this paper, assuming that each node is incident with two or more fault-free links, we show that an n-dimensional alternating group graph can tolerate up to 4n − 13 link faults, where n ? 4, while retaining a fault-free Hamiltonian cycle. The proof is computer-assisted. The result is optimal with respect to the number of link faults tolerated. Previously, without the assumption, at most 2n − 6 link faults can be tolerated for the same problem and the same graph.     

Strong Menger connectivity with conditional faults on the class of hypercube-like networks

In this paper, we study the Menger property on a class of hypercube-like networks. We show that in all n-dimensional hypercube-like networks with n−2 vertices removed, every pair of unremoved vertices u and v are connected by min{deg(u),deg(v)} vertex-disjoint paths, where deg(u) and deg(v) are the remaining degree of vertices u and v, respectively. Furthermore, under the restricted condition that each vertex has at least two fault-free adjacent vertices, all hypercube-like networks still have the strong Menger property, even if there are up to 2n−5 vertex faults.     

The Cayley network of expanded Pancake graphs

Signed permutation group has important applications in genome rearrangement as well as the construction of networks. In this paper, we propose a new interconnection network named extended Pancake graph, we investigate its topological properties, and give a routing algorithm with the diameter upper bounded by 2n?1. Some embedding properties are also derived. In conclusion, we present a comparison of some familiar networks with the Cayley graph EP _n .