Information Sciences 375 (2017) 258–270 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins Hamiltonian paths in hypercubes with local traps Janusz Dybizba ´ nski ∗ , Andrzej Szepietowski Institute of Informatics, Faculty of Mathematics, Physics and Informatics, University of Gda´ nsk, 80–308 Gda´ nsk, Poland a r t i c l e i n f o Article history: Received 19 January 2016 Revised 21 April 2016 Accepted 3 October 2016 Available online 13 October 2016 Keywords: Hamiltonian path Hypercube Fault tolerance a b s t r a c t The n-dimensional hypercube Q n is a graph with 2 n vertices, each labeled with a distinct binary string of length n. The vertices are connected by an edge if and only if their labels differ in one bit. The hypercube is bipartite, the set of nodes is the union of two sets: nodes of parity 0 (the number of ones in their labels is even) and nodes of parity 1 (the number of ones is odd). We consider Hamiltonian paths in hypercubes with faulty edges and prove the following: (1) If Q n has one vertex u of degree 1, then u can be connected by a Hamiltonian path with every vertex v that is of a parity different than u and that is not connected with u by a healthy edge. (2) If Q n with n ≥ 4 has two vertices u and v of degree 1, then they can be connected by a Hamiltonian path if the distance between u and v is odd and greater than 1 or if u and v are connected by the faulty edge. (3) If Q n contains a cycle (u, v, w, x) in which all edges going away from the cycle from u and w are faulty, then u or w can be connected by a Hamiltonian path with any vertex outside the cycle that is of different parity than u and w. Moreover, in all three cases, the thesis remains true even if Q n has n − 3 additional faulty edges. Furthermore, in all three cases, no other Hamiltonian paths are possible. © 2016 Elsevier Inc. All rights reserved. 1. Introduction An n-dimensional hypercube (n-cube), denoted by Q n , is a graph with 2 n vertices, each labeled with a distinct binary string of length n. We identify the vertices of Q n with integers in {0, . . . , 2 n − 1} in the usual way; see Figs. 3 or 5. Two vertices are connected by an edge if and only if their labels differ in one bit. The hypercube is one of the most popular interconnecting networks. It has several excellent properties, such as a recursive structure, symmetry, small diameter, low degree and easy routing algorithm, which are important when designing parallel or distributed systems. The hypercube Q n is bipartite, the set of nodes is the union of two sets: nodes of parity 0 (the number of ones in their labels is even) and nodes of parity 1 (the number of ones is odd), and each edge connects nodes of different parity. A path or a cycle is Hamiltonian if it visits each node in the cube exactly once. The hypercube is Hamiltonian, i.e., it contains a Hamiltonian cycle. Moreover, it is laceable, which means that any two vertices of different parity can be connected by a Hamiltonian path. Note that the ends of every Hamiltonian path are of different parity because the cube Q n is bipartite. Hence, vertices of the same parity cannot be connected by a Hamiltonian path. The existence of Hamiltonian cycles or paths in interconnection networks is useful, particularly for parallel and distributed computing. Some fundamental, simple and important parallel algorithms are designed for processors arranged in a path or a ring; see [14,19]. In addition, the Hamiltonian property is fundamental for the deadlock-free routing algorithms [20,27,28]. An important property of the hypercube is its fault tolerance. It contains ∗ Corresponding author: Fax: +48585232158. E-mail addresses: jdybiz@inf.ug.edu.pl (J. Dybizba ´ nski), matszp@inf.ug.edu.pl (A. Szepietowski). http://dx.doi.org/10.1016/j.ins.2016.10.011 0020-0255/© 2016 Elsevier Inc. All rights reserved.