Hamiltonian cycles in hypercubes with faulty edges

Abstract

Szepietowski [A. Szepietowski, Hamiltonian cycles in hypercubes with $2n-4$ faulty edges, Information Sciences, 215 (2012) 75--82] observed that the hypercube $Q_n$ is not Hamiltonian if it contains a trap disconnected halfway. A proper subgraph $T$ is disconnected halfway if at least half of its nodes have parity 0 (or 1, resp.) and the edges joining all nodes of parity 0 (or 1, resp.) in $T$ with nodes outside $T$, are faulty. The simplest examples of such traps are: (1) a vertex with $n-1$ incident faulty edges, or (2) a cycle $(u,v,w,x)$, where all edges going out of the cycle from $u$ and $w$ are faulty. In this paper we describe all traps disconnected halfway $T$ with the size $|T|\le8$, and discuss the problem whether there exist small sets of faulty edges which preclude Hamiltonian cycles and are not based on sets disconnected halfway. We describe heuristic which detects sets of faulty edges which preclude HC also those sets that are not based on subgraphs disconnected halfway. We describe all $Q_4$ cubes that are not Hamiltonian, and all $Q_5$ cubes with 8 or 9 faulty edges that are not Hamiltonian.