Edge-bipancyclicity of star graphs under edge-fault tolerant

Abstract

The star graph S_n is one of the most famous interconnection networks. It has been shown by Li [T.-K. Li, Cycle embedding in star graphs with edge faults, Appl. Math. Comput. 167 (2005) 891–900] that S_n contains a cycle of length from 6 to n! when the number of fault edges in the graph does not exceed n − 3. In this paper, we improve this result by showing that for any edge subset F of S_n with ∣F∣ ⩽ n − 3 every edge of S_n − F lies on a cycle of every even length from 6 to n! provided n ⩾ 3.

Introduction

In interconnection networks, the problem of simulating one network by another is modelled as a graph embedding problem. There are several reasons why such an embedding is important [11]. For example, there are a number of efficient algorithms for solving some application problems and best communication patterns for their executions. For these algorithms, the existence of certain topological structures guarantee the desired performance. Thus, for such applications, it is desired to provide logically a specific topological structure throughout the execution of the algorithm in the network design.

Among all embedding problems, cycle embedding problem is one of the most popular problems, that is, finding a cycle of given length in a graph. A graph G is called pancyclic [3] if there exists a cycle of every length from 3 to ∣V(G)∣. A graph is bipartite graph if its vertex-set can be partitioned into two disjoint subsets such that each edge is incident to two vertices from different subsets. A bipartite graph G is called bipancyclic if there exists a cycle of every even length from 4 to ∣V(G)∣. The pancyclicity is an important metric in embedding cycles of any length into the topology of network. The concept of pancyclicity was extended to vertex-pancyclicity by Hobbs [6] and edge-pancyclicity by Alspach and Hare [2]. A graph G is called vertex-pancyclic if for any vertex u, there exists a cycle of every length from 3 to ∣V(G)∣ containing u; and edge-pancyclic if for any edge e, there exists a cycle of every length from 3 to ∣V(G)∣ containing e. Obviously, every edge-pancyclic graph is vertex-pancyclic. A bipartite graph G is vertex-bipancyclic if for any vertex u, there exists a cycle of every even length from 4 to ∣V(G)∣ containing u. Similarly, a bipartite graph G is called edge-bipancyclic if for any edge e, there exists a cycle of every even length from 4 to ∣V(G)∣ containing e. A graph G is said to be Hamiltonian connected if there exists a Hamiltonian path between any two vertices of G. It is easy to see that any bipartite graph with at least three vertices is not Hamiltonian connected. For this reason, Simmons [10] introduced the concept of Hamiltonian laceable for Hamiltonian bipartite graphs. A Hamiltonian bipartite graph is Hamiltonian laceable if there is a Hamiltonian path between any two vertices in different bipartite sets. Obviously, a Hamilton cycle can be embedded in the Hamiltonian connected graphs. Then the Hamiltonian connectivity is also important metric in embedding Hamitonian cycles into the topology of network. Since some components in a network would sometimes fail, it’s more practical to study graphs with faults.

Star graphs, proposed by Akers and Krishnamurthy [1], is a famous interconnection networks. In this paper, we explore the embedding problems on star graphs. They proved that the star graphs are Cayley graphs, thus they are vertex symmetric. Furthermore, the star graphs have many other nice properties such as recursiveness, edge-symmetry [1]. Since the star graphs are bipartite graphs, odd cycles cannot be embedded into it. Jwo et al. [7] showed that any cycle of even length from 6 to n! can be embedded into S_n. Hsieh et al. [5] and Li et al. [9], proved that the n-dimensional star graph S_n is (n − 3)-edge-fault tolerant Hamiltonian laceable for n ⩾ 4. Recently, Li [8] considered the edge-fault tolerance of star graphs and showed that cycles of even length from 6 to n! can be embedded into the n-dimensional star graphs when the number of the fault edges are less than n − 3. In this paper, we improve this result by showing that for any edge subset F of S_n with ∣F∣ ⩽ n − 3 and any edge e ∈ S_n − F, there exists a cycle of even length from 6 to n! in S_n − F containing e provided n ⩾ 3.

The rest of this paper is organized as follows. In Section 2, we give the definition and basic properties of the n-dimensional star graph S_n. In Section 3, we discuss the edge-fault-tolerant edge-bipancyclicity of the star graphs.