Fault-Tolerant Embedding of Pairwise Independent Hamiltonian Paths on a Faulty Hypercube with Edge Faults

Abstract

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.