Edge-fault-tolerant pancyclicity of arrangement graphs

Abstract

The arrangement graph $A_{n\text{,}k}$ is a well-known interconnection network. Day and Tripathi proved that $A_{n\text{,}k}$ is pancyclic for $n-k⩾2$. In this paper, we improve this result, and we demonstrate that $A_{n\text{,}k}$ is also pancyclic even if it has no more than $(k(n-k)-2)$ faulty edges for $n-k⩾2$. Our result is optimal concerning the edge fault-tolerance.

Introduction

The interconnection network is an important research area for parallel and distributed computer systems. Usually, the topology of a network can be represented as a graph in which the vertices represent processors and the edges represent the communication links.

The star graph, which was proposed by Akers et al. [1], is a well-known interconnection network. As a generalization of the star graph, Day and Tripathi [5] proposed the arrangement graph. For a positive integer s, let $〈s〉$ denote the set $\{1\text{,}2\text{,}\dots \text{,}s\}$. Given two positive integers n and k with $n>k⩾1$, the $(n\text{,}k)$-arrangement graph $A_{n\text{,}k}$ is the graph that has the vertex set $V(A_{n\text{,}k})=\{u=u_1{u}_2\dots u_k|u_i\in 〈n〉\text{,}{u}_i\ne u_j\mathrm{if}i\ne j\}$ and the edge set $E(A_{n\text{,}k})=\{(p\text{,}q)|p\text{,}q\in V(A_{n\text{,}k})\text{,}\mathrm{and}p\text{,}q\mathrm{differ}\mathrm{in}$ exactly one position}. From the definition, we know that $A_{n\text{,}k}$ is a regular graph of degree $k(n-k)$ with $\frac{n!}{(n-k)!}$ vertices, $A_{n\text{,}1}$ is isomorphic to the complete graph $K_n$, and $A_{n\text{,}n-1}$ is isomorphic to the n-dimensional star graph. Moreover, $A_{n\text{,}k}$ is vertex-transitive and edge-transitive [5]. Fig. 1 shows the arrangement graph $A_{4\text{,}2}$.

For $i\in 〈n〉\text{,}j\in 〈k〉$, suppose that $A_{n\text{,}k}^{(j:i)}$ denotes the subgraph of $A_{n\text{,}k}$ that is induced by $V\left(A_{n\text{,}k}^{(j:i)}\right)=\{p|p=p_1{p}_2\dots p_k\mathrm{and}{p}_j=i\}$. Obviously, $\{V(A_{n\text{,}k}^{(j:i)})|1⩽i⩽n\}$ forms a partition of $V(A_{n\text{,}k})$ and each $A_{n\text{,}k}^{(j:i)}$ is isomorphic to $A_{n-1\text{,}k-1}$. Then $A_{n\text{,}k}$ can be recursively constructed from n copies of $A_{n-1\text{,}k-1}$ and every two copies have $\frac{(n-2)!}{(n-k-1)!}$ edges between them. We follow [4] for graph-theoretical terminologies and notations. For two paths $P=〈u_0\text{,}{u}_1\text{,}\dots \text{,}{u}_m〉$ and $P^{\prime }=〈u_m\text{,}{u}_{m+1}\text{,}\dots \text{,}{u}_n〉\text{,}P+P^{\prime }$ denote the path $〈u_0\text{,}{u}_1\text{,}\dots \text{,}{u}_m\text{,}{u}_{m+1}\text{,}\dots \text{,}{u}_n〉$.

In interconnection networks, the problem of simulating one network by another is modeled as a graph-embedding problem. There are several reasons why such an embedding is important [24]. For example, the execution of an efficient algorithm requires certain topological structures. Thus, it is desired to provide logically a specific topological structure throughout the execution of the algorithm in the network design.

The cycle embedding problem is one of the most popular embedding problems. This problem is to find a cycle of given length in a graph. A graph G is Hamiltonian if it has a cycle of length $|V(G)|$. If there exists a Hamiltonian path (a path of length $|V(G)|-1$) between any two vertices of G, then the graph G is said to be Hamiltonian connected. A graph G is pancyclic if it has cycles of each length from g to $|V(G)|$ where g is the length of a shortest cycle of G. Let $d_G(u\text{,}v)$ denote the length of a shortest path between vertices u and v in G. A graph G is panconnected if for any two vertices $x\text{,}y$ in G, there exist paths between x and y of each length from $d_G(x\text{,}y)$ to $|V(G)|-1$. The pancyclicity is an important metric when embedding cycles of any length into the topology of a network. A large amount of related work has appeared in the literature [2], [3], [6], [13], [14], [15], [23].

Because some components in a graph could fail sometimes, it is more practical to study graphs with faults. A graph G is k-fault-tolerant pancyclic (resp. Hamiltonian, Hamiltonian connected, panconnected) if $G-F$ remains pancyclic (resp. Hamiltonian, Hamiltonian connected, panconnected) for $F\subseteq V(G)\cup E(G)\text{,}|F|⩽k$. A graph G is k-edge-fault-tolerant pancyclic (resp. Hamiltonian, Hamiltonian connected, panconnected) if $G-F$ remains pancyclic (resp. Hamiltonian, Hamiltonian connected, panconnected) for $F\subseteq E(G)\text{,}|F|⩽k$. The fault-tolerant pancyclicity has been investigated widely. There is a substantial amount of related literature [7], [8], [9], [10], [11], [12], [15], [16], [19], [20], [21], [22], [25].

For $n-k=1$, the cycle embedding problems of the arrangement graph $A_{n\text{,}k}$, which is a star graph, have been discussed in [7], [13], [15], [21], [25]. For $n-k⩾2$, Day and Tripathi [6] proved that $A_{n\text{,}k}$ is pancyclic. Teng et al. [18] proved that $A_{n\text{,}k}$ is panconnected. Concerning fault tolerance, Hsieh et al. [10] studied the existence of Hamiltonian cycles in faulty arrangement graphs. Hsu et al. [12] proved that $A_{n\text{,}k}$ is $(k(n-k)-2)$-fault-tolerant Hamiltonian if $n-k⩾2$. Lo and Chen [17] proved that $A_{n\text{,}k}$ is $(k(n-k)-2)$-edge-fault-tolerant Hamiltonian connected if all faulty edges are not adjacent to the same vertex. In this paper, we prove that $A_{n\text{,}k}$ is $(k(n-k)-2)$-edge-fault-tolerant pancyclic if $n-k⩾2$. If there are $k(n-k)-1$ faulty edges and all of them are adjacent to the same vertex in $A_{n\text{,}k}$, then $A_{n\text{,}k}-F$ is not Hamiltonian. This finding demonstrates that our result is optimal with respect to edge fault-tolerance.