Fault-tolerant cycle-embedding in alternating group graphs
Introduction
Interconnection networks are usually modeled as undirected simple graphs G = (V, E), where the vertex set V = V(G) represents the set of processing elements and the edge set E = E(G) represents the set of communication channels, respectively. One of the central issues in evaluating the efficiency of interconnection networks is to consider the ability of graph embedding [3], [4]. Especially, cycle-embedding is the most important issue because networks with cycle topology are suitable for designing simple algorithms with low communication costs. In addition, the study of cycle-embedding in networks can also be viewed as an extension of theoretical research on Hamiltonicity. Recently, the topic has attracted a burst of investigations (see, for example, [1], [2], [9], [10], [11], [13], [21], [22], [25]). Moreover, cycle-embedding is also concerned extensively in many diverse interconnection networks with faulty elements [12], [15], [16], [17], [19], [20], [23], [24], [26].
A path connecting two vertices u and v in a graph G is called a u–v path. The distance between u and v, denoted by dG(u, v), is the number of edges in a shortest u–v path. A path (respectively, cycle) that contains every vertex of a graph exactly once is called a Hamiltonian path (respectively, Hamiltonian cycle). This paper is concerned with the following Hamiltonian-like properties. A graph G is traceable (respectively, Hamiltonian) if it possesses a Hamiltonian path (respectively, a Hamiltonian cycle). A graph G is Hamiltonian-connected if every two vertices of G are connected by a Hamiltonian path. For an integer r ⩾ 3, a graph G is called r-pancyclic if it contains a cycle of length ℓ (i.e., an ℓ-cycle) for each ℓ with r ⩽ ℓ ⩽ ∣V∣. In particular, G is called vertex r-pancyclic (respectively, edge r-pancyclic) if every vertex (respectively, edge) of G belongs to an ℓ-cycle for each ℓ with r ⩽ ℓ ⩽ ∣V∣. A 3-pancyclic graph, a vertex 3-pancyclic graph, and an edge 3-pancyclic graph are called pancyclic, vertex-pancyclic, and edge-pancyclic, respectively. A graph G is panconnected if, for any two distinct vertices u, v ∈ V and for each integer ℓ with dG(u, v) ⩽ ℓ ⩽ ∣V∣ − 1, there is a u–v path of length ℓ in G. The existence of pancyclicity in a network is even more important because it implies that the network can embed cycles with arbitrary length.
In this paper, we study the fault-tolerant pancyclicity for a particular family of interconnection networks called alternating group graphs. Alternating group graphs were first proposed by Jwo et al. [14] and further investigated by Cheng and Lipman [6], [7], [8]. This family of graphs has been shown to have many nice properties including vertex-transitivity, edge-transitivity, strongly hierarchy, maximally connectivity (i.e., the connectivity equals to the regularity in a graph), and has small diameter and average distance. Moreover, alternating group graphs have many advantages over the families of hypercubes and star graphs. For example, alternating group graphs possess stronger properties on Hamiltonicity (e.g., the graphs are not only Hamiltonian-connected and pancyclic [14] but also panconnected [5]). Let F be a set of faulty elements in a graph G and G − F denotes the residual graph of G by removing the faulty elements. For n-dimensional alternating group graph AGn with n ⩾ 4 and F ⊂ V(AGn), Chang et al. [5] proved that AGn − F is Hamiltonian if ∣F∣ ⩽ n − 2 and is Hamiltonian-connected if ∣F∣ ⩽ n − 3. In this paper, we show that, for n ⩾ 4 and F ⊂ V(AGn), the following fault-tolerant properties hold: (i) AGn − F is pancyclic if ∣F∣ ⩽ n − 2; (ii) AGn − F is vertex-pancyclic if ∣F∣ ⩽ n − 3; and (iii) AGn − F is edge 4-pancyclic if ∣F∣ ⩽ n − 4. The first case leads to an improvement over the result of [5] and the last two cases obtain new characterizations of fault-tolerant pancyclicity on AGn.
The rest of this paper is organized as follows. In Section 2, we formally define the alternating group graphs AGn and present some basic properties of them. In Section 3, we give the proofs of our main results. Finally, some concluding remarks are contained in Section 4.
Section snippets
Structural properties of AGn
This section is devoted to introducing the structural properties of alternating group graphs. Define 〈n〉 as the set {1, 2, … , n} and let p = p1p2 ⋯ pn be a permutation of the elements of 〈n〉, i.e., pi ∈ 〈n〉 and pi ≠ pj for i ≠ j. A pair of elements pi and pj is said to be an inversion of p if pi < pj whenever i > j. An even permutation is a permutation that contains an even number of inversions. Let An denote the set of all even permutations over 〈n〉 and let gij be an operation on An that swaps elements in
Fault-tolerant pancyclicity
Before stating the main result of this section, we first note that if p ∈ F is a faulty vertex in AGn, then every edge for i = 3, 4, … , n cannot lie on a 3-cycle in AGn − F. Thus, we may only consider edge r-pancyclicity for r ⩾ 4 in the residual graph. Theorem 1 For n ⩾ 4, AGn is (n − 4)-fault-tolerant edge 4-pancyclic. Proof We prove the theorem by induction on n ⩾ 4. By Lemma 1, the result holds for AG4 since every panconnected graph is edge-pancyclic (and thus is also edge 4-pancyclic). Suppose that n ⩾ 5 and the
Concluding remarks
We close this paper with the following conclusions. We fist point out that all the results of fault-tolerant pancyclicity in this paper are the best possible in the sense that the number of faulty vertices cannot be increased. For instance, if AG4 has three faulty vertices adjacent to a faultless vertex, then the remaining graph contains no Hamiltonian cycle and thus is not pancyclic. Also, if F = {3124, 4132}, then the vertex 1234 cannot lie on any 3-cycle in AG4 − F. And further, if F = {2431}, then
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