Hamiltonian cycles in hypercubes with faulty edges

TL;DR

This paper characterizes small traps that prevent Hamiltonian cycles in hypercubes with faulty edges, and develops heuristics to detect such faults beyond known trap structures.

Contribution

It provides a complete description of small traps disconnected halfway in hypercubes and introduces heuristics to identify faulty edge sets that block Hamiltonian cycles.

Findings

All traps disconnected halfway with size ≤8 are described.

All non-Hamiltonian Q4 hypercubes are characterized.

Non-Hamiltonian Q5 hypercubes with 8 or 9 faulty edges are identified.

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.