Fault-free Hamiltonian cycles passing through a prescribed linear forest in 3-ary n-cube with faulty edges

Abstract

The k-ary n-cube Qnk (n ⩾ 2 and k ⩾ 3) is one of the most popular interconnection networks. In this paper, we consider the problem of a faultfree Hamiltonian cycle passing through a prescribed linear forest (i.e., pairwise vertex-disjoint paths) in the 3-ary n-cube Qn3 with faulty edges. The following result is obtained. Let E0 (≠ ∅) be a linear forest and F (≠= ∅) be a set of faulty edges in Qn3 such that E0 ∩ F = ∅ and |E0| + |F| ⩽ 2n − 2. Then all edges of E0 lie on a Hamiltonian cycle in Qn3 − F, and the upper bound 2n − 2 is sharp.