Panpositionable Hamiltonicity of the Alternating Group Graphs Yuan-Hsiang Teng and Jimmy J. M. Tan Department of Computer Science, National Chiao Tung University, Hsinchu City, Taiwan 300, Republic of China Lih-Hsing Hsu Department of Computer Science and Information Engineering, Providence University, Taichung County, Taiwan 433, Republic of China The alternating group graph AG n is an interconnection network topology based on the Cayley graph of the alternating group. There are some interesting results concerning the hamiltonicity and the fault tolerant hamil- tonicity of the alternating group graphs. In this article, we propose a new concept called panpositionable hamil- tonicity. A hamiltonian graph G is panpositionable if for any two different vertices x and y of G and for any integer l satisfying d (x , y ) ≤ l ≤|V (G)|- d (x , y ), there exists a hamiltonian cycle C of G such that the relative distance between x , y on C is l . We show that AG n is panposi- tionable hamiltonian if n ≥ 3. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(2), 146–156 2007 Keywords: alternating group graph; hamiltonian; hamiltonian connected; panpositionable hamiltonian 1. INTRODUCTION Network topology is usually represented by a graph where the vertices represent processors and the edges represent the links between processors. For graph definitions and nota- tion we follow Ref. [6]. Let G = (V , E) be a graph, where 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 is the vertex set and E is the edge set of G. Two vertices u and v are adjacent if (u, v) ∈ E. A path P is represented by 〈v 0 , v 1 , v 2 , ... , v k 〉. The length of a path P is the number of edges in P, denoted by L(P). We sometimes write the path 〈v 0 , v 1 , v 2 , ... , v k 〉 as 〈v 0 , P 1 , v i , v i+1 , ... , v j , P 2 , v t , ... , v k 〉, where P 1 is the path 〈v 0 , v 1 , ... , v i 〉 and P 2 is the path 〈v j , v j+1 , ... , v t 〉. It is Received January 2006; accepted November 2006 Correspondence to: J. M. Tan; e-mail: jmtan@cs.nctu.edu.tw Contract grant sponsor: National Science Council of the Republic of China; Contract grant number: NSC 94-2213-E-009-138 DOI 10.1002/net.20184 Published online in Wiley InterScience (www.interscience.wiley. com). © 2007 Wiley Periodicals, Inc. possible to write a path 〈v 0 , v 1 , P, v 1 , v 2 , ... , v k 〉 if L(P) = 0. We use d G (u, v), or simply d (u, v) if there is no ambiguity, to denote the distance between u and v in a graph G, i.e., the length of a shortest path joining u and v in G. A cycle is a path with at least three vertices such that the first vertex is the same as the last one. We use d C (u, v) and D C (u, v) to denote the shorter and the longer distance between u and v on a cycle C of G, respectively. It is possible that D C (u, v) = d C (u, v) if the lengths of the two disjoint paths joining u and v in C are equal. A path is a hamiltonian path if its vertices are distinct and span V . A graph G is hamiltonian connected if there exists a hamiltonian path joining any two vertices of G. A hamiltonian cycle of G is a cycle that traverses every vertex of G exactly once. A graph G is hamiltonian if there exists a hamiltonian cycle in G. The hamiltonian properties are impor- tant aspects of designing an interconnection network. Many related works have appeared in the literature [1, 5, 7]. We propose a new concept called panpositionable hamil- tonicity. A hamiltonian graph G is panpositionable if for any two different vertices x and y of G and for any integer l satisfy- ing d (x, y) ≤ l ≤|V (G)|− d (x, y), there exists a hamiltonian cycle C of G such that the relative distance between x, y on C is l ; more precisely, d C (x, y) = l if l ≤⌊ |V (G)| 2 ⌋ or D C (x, y) = l if l > |V (G)| 2 . Given a hamiltonian cycle C, if d C (x, y) = l , we have D C (x, y) =|V (G)|− d C (x, y). There- fore, a graph is panpositionable hamiltonian if for any integer l with d (x, y) ≤ l ≤ |V (G)| 2 , there exists a hamiltonian cycle C of G with d C (x, y) = l . One trivial example, the complete graph K n with n ≥ 3, is panpositionable. There are several requirements in designing a good topol- ogy for an interconnection network, such as connectivity and hamiltonicity. The hamiltonian property is one of the major requirements in designing an interconnection network. The hamiltonian property is fundamental to the deadlock-free routing algorithms of distributed systems [8, 9]. A high- reliability network design can be based on constructing a hamiltonian cycle in an interconnection network. Similar NETWORKS—2007—DOI 10.1002/net