- Published 2008 in Theory of Computing Systems

A Hamiltonian path in G is a path which contains every vertex of G exactly once. Two Hamiltonian paths P 1=〈u 1,u 2,…,u n 〉 and P 2=〈v 1,v 2,…,v n 〉 of G are said to be independent if u 1=v 1, u n =v n , and u i ≠v i for all 1<i<n; and both are full-independent if u i ≠v i for all 1≤i≤n. Moreover, P 1 and P 2 are independent starting at u 1, if u 1=v 1 and u i ≠v i for all 1<i≤n. A set of Hamiltonian paths {P 1,P 2,…,P k } of G are pairwise independent (respectively, pairwise full-independent, pairwise independent starting at u 1) if any two different Hamiltonian paths in the set are independent (respectively, full-independent, independent starting at u 1). A bipartite graph G is Hamiltonian-laceable if there exists a Hamiltonian path between any two vertices from different partite sets. It is well known that an n-dimensional hypercube Q n is bipartite with two partite sets of equal size. Let F be the set of faulty edges of Q n . In this paper, we show the following results: 1. When |F|≤n−4, Q n −F−{x,y} remains Hamiltonian-laceable, where x and y are any two vertices from different partite sets and n≥4. 2. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise full-independent Hamiltonian paths between n−|F|−1 pairs of adjacent vertices, where n≥2. 3. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths starting at any vertex v in a partite set to n−|F|−1 distinct vertices in the other partite set, where n≥2. 4. When 1≤|F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths between any two vertices from different partite sets, where n≥3. When |F|≤n−4, Q n −F−{x,y} remains Hamiltonian-laceable, where x and y are any two vertices from different partite sets and n≥4. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise full-independent Hamiltonian paths between n−|F|−1 pairs of adjacent vertices, where n≥2. When |F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths starting at any vertex v in a partite set to n−|F|−1 distinct vertices in the other partite set, where n≥2. When 1≤|F|≤n−2, Q n −F contains (n−|F|−1)-pairwise independent Hamiltonian paths between any two vertices from different partite sets, where n≥3.

@article{Hsieh2008FaultTolerantEO,
title={Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults},
author={Sun-Yuan Hsieh and Yu-Fen Weng},
journal={Theory of Computing Systems},
year={2008},
volume={45},
pages={407-425}
}