View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 55 (2008) 2076–2085 www.elsevier.com/locate/camwa Hyper-Hamiltonian generalized Petersen graphs Ta-Cheng Maia, Jeng-Jung Wanga,∗, Lih-Hsing Hsub a Department of Information Engineering, I-Shou University, Kaohsiung, 84008 Taiwan, ROC b Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan, ROC Received 8 November 2006; received in revised form 7 June 2007; accepted 30 August 2007 Abstract Assume that n and k are positive integers with n ≥ 2k + 1. A non-Hamiltonian graph G is hypo-Hamiltonian if G − v is Hamiltonian for any v ∈ V (G). It is proved that the generalized Petersen graph P(n, k) is hypo-Hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). Similarly, a Hamiltonian graph G is hyper-Hamiltonian if G − v is Hamiltonian for any v ∈ V (G). In this paper, we will give some necessary conditions and some sufficient conditions for the hyper-Hamiltonian generalized Petersen graphs. In particular, P(n, k) is not hyper-Hamiltonian if n is even and k is odd. We also prove that P(3k, k) is hyper-Hamiltonian if and only if k is odd. Moreover, P(n, 3) is hyper-Hamiltonian if and only if n is odd and P(n, 4) is hyper-Hamiltonian if and only if n 6= 12. Furthermore, P(n, k) is hyper-Hamiltonian if k is even with k ≥ 6 and n ≥ 2k + 2 + (4k − 1)(4k + 1), and P(n, k) is hyper-Hamiltonian if k ≥ 5 is odd and n is odd with n ≥ 6k − 3 + 2k(6k − 2). c 2007 Elsevier Ltd. All rights reserved. Keywords: Hamiltonian; Hypo-Hamiltonian; Hyper-Hamiltonian; Lattice diagram; Generalized Petersen graph 1. Definitions and notations For the graph terminology and notation we follow [1]. G = (V, E) is a graph if V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V (G) is the vertex set and E(G) is the edge set of G. We delimited a path P in a graph from a vertex v0 to vn by hv0, v1, v2, . , vni. A cycle is a path with at least three vertices such that its first vertex is the same as the last vertex. A cycle is a Hamiltonian cycle if it traverses every vertex of G exactly once. A graph is Hamiltonian if it has a Hamiltonian cycle. Assume that n and k are positive integers with n ≥ 2k + 1. We use ⊕ to denote addition in integer modular n. The generalized Petersen graph P(n, k) is the graph with vertex set {i | 0 ≤ i ≤ n − 1} ∪ {i0 | 0 ≤ i ≤ n − 1} and edge set {(i, i ⊕ 1) | 0 ≤ i ≤ n − 1} ∪ {(i, i0) | 0 ≤ i ≤ n − 1} ∪ {(i0,(i ⊕ k)0) | 0 ≤ i ≤ n − 1}. The Hamiltonian generalized Petersen graph has been extensively studied [2–6]. In particular, Alspach [3] gave the classification on Hamiltonian generalized Petersen graphs as follows. Theorem 1 ([3]). P(n, k) is Hamiltonian if and only if it is neither (1) P(n, 2) =∼ P(n, n − 2) =∼ P(n,(n − 1)/2) =∼ P(n,(n + 1)/2), n ≡ 5 (mod 6) nor (2) P(n, n/2), n ≡ 0 (mod 4) and n ≥ 8. ∗ Corresponding author. E-mail address: [email protected] (J.-J. Wang). 0898-1221/$ - see front matter c 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2007.08.041 T.-C. Mai et al. / Computers and Mathematics with Applications 55 (2008) 2076–2085 2077 In this paper we concentrate on cubic generalized Petersen graphs, the case k = n/2 will be exclusive for consideration. Therefore P(n, k) is not Hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). However, all non- Hamiltonian generalized Petersen graphs satisfy another interesting property. A non-Hamiltonian graph G such that G − v is Hamiltonian for any v ∈ V (G) is called a hypo-Hamiltonian graph. It is proved that the set of hypo- Hamiltonian generalized Petersen graphs is actually the set of non-Hamiltonian generalized Petersen graphs [5]. Similarly, a Hamiltonian graph G is hyper-Hamiltonian if G − v is Hamiltonian for any v ∈ V (G). We are interested in the recognition of hyper-Hamiltonian generalized Petersen graphs. In [7], it is proved that P(n, 1) is hyper-Hamiltonian if and only if n is odd and P(n, 2) is hyper-Hamiltonian if and only if n ≡ 1, 3 (mod 6). In this paper, we will give some necessary conditions and some sufficient conditions for the hyper-Hamiltonian generalized Petersen graphs. In particular, P(n, k) is not hyper-Hamiltonian if n is even and k is odd. We also proved that P(3k, k) is hyper-Hamiltonian if and only if k is odd. Moreover, P(n, 3) is hyper-Hamiltonian if and only if n is odd; P(n, 4) is hyper-Hamiltonian if and only if n 6= 12. Furthermore, P(n, k) is hyper-Hamiltonian if k is even with k ≥ 6 and n ≥ 2k + 2 + (4k − 1)(4k + 1), and P(n, k) is hyper-Hamiltonian if k is odd with k ≥ 5 and n is odd with n ≥ 6k − 3 + 2k(6k − 2). Alspach et al. [2] proposed an interesting model, called lattice model, to describe a Hamiltonian cycle for given generalized Petersen graph. With the lattice model, a lattice diagram for a generalized Petersen graphs is a labeled graph in the (x, y)-plane that possesses a closed or an open Eulerian trail. By appropriately interpreting the edges in the diagram, the Eulerian trail corresponds to a Hamiltonian cycle of a generalized Petersen graph. Based on this model, they proved that P(n, k) is Hamiltonian if k ≥ 3 and n sufficiently large [2] and moreover classified the Hamiltonian generalized Petersen graphs [3]. The lattice model is described as follows. For more details, see [2]. In lattice model, a labeled lattice graph L(n, k) consists of lattice points in the (x, y)-plane. If a lattice point (a, b) is labeled with an integer i with 0 ≤ i ≤ n − 1, then (a + 1, b) is labeled with i ⊕ 1 and (a, b − 1) is labeled with i ⊕ k. A lattice diagram for P(n, k), denoted as D(n, k), is a subgraph of L(n, k) induced by the vertices with labels 0, 1,..., n − 1 such that it possesses either a closed or an open Eulerian trail. A traversal of the Eulerian trail in D(n, k) obeys the following rules: 1. The trail does not change the direction when it passes through a vertex of degree 4. 2. Each label 0, 1,..., n − 1 is encountered by the traversal once in a vertical direction and once in a horizontal direction. 3. If D(n, k) has an open Eulerian trail, then the two vertices of odd degree must have the same label and not both being of degree 3. The correspondence between an Eulerian trail of a lattice diagram D(n, k) and a Hamiltonian cycle of the generalized Petersen graph P(n, k) is built by interpreting the edges in L(n, k). The interpretations of edges in L(n, k) are given as follows: 1. The vertical edge (i, i ⊕ k) in L(n, k) corresponds to an edge (i0,(i ⊕ k)0) in P(n, k). 2. The horizontal edge (i, i ⊕ 1) in L(n, k) corresponds to an edge (i, i ⊕ 1) in P(n, k). 3. Two edges in different directions incident to the vertex i of degree 2 in L(n, k) correspond to an edge (i, i0) in P(n, k). A lattice diagram D(13, 4) for a generalized Petersen P(13, 4), for example, is shown in Fig. 1. In this diagram, there is an Eulerian trail h0, 1, 5, 9, 10, 6, 2, 3, 7, 11, 12, 8, 7, 6, 5, 4, 0i. By appropriately interpreting the edges in the diagram, it corresponds to a Hamiltonian cycle h0, 1, 10, 50, 90, 9, 10, 100, 60, 20, 2, 3, 30, 70, 110, 11, 12, 120, 80, 8, 7, 6, 5, 4, 40, 00, 0i in P(13, 4). It is not difficult to see that an Eulerian trail in L(n, k) corresponds to a Hamiltonian cycle of P(n, k) and any Hamiltonian cycle of P(n, k) can be converted into an Eulerian trail in D(n, k). Hence, finding a Hamiltonian cycle of a generalized Petersen graph P(n, k) is finding an appropriate lattice diagram for P(n, k) [2]. In addition, Alspach et al. [2] proposed an amalgamating mechanism to generate lattice diagrams with various sizes. For example, we have a lattice diagram D(4k, k) in Fig. 2(a) and a lattice diagram D(4k + 2, k) in Fig. 2(b). By identifying the vertex with label (4k − 1) of D(4k, k) and the vertex with label 0 of D(4k + 2, k) and relabeling all vertices of D(4k + 2, k) by adding (4k − 1) to all the labels, we obtain a lattice diagram D(8k + 1, k) shown in Fig. 2(c). Thus, P(8k + 1, k) is Hamiltonian. With this mechanism, we can amalgamate r copies of D(4k, k) and s copies of D(4k + 2, k), where r ≥ 1 and s ≥ 0, to make a lattice diagram D(4k + (r − 1)(4k − 1) + s(4k + 1), k).