A fault-free Hamiltonian cycle passing through prescribed edges in a hypercube with faulty edges

Abstract

In this paper, we consider the problem of a fault-free Hamiltonian cycle passing through prescribed edges in an n-dimensional hypercube Q n with some faulty edges. We obtain the following result: Let n ⩾ 2 , F ⊂ E ( Q n ) , E 0 ⊂ E ( Q n ) \\ F with 1 ⩽ | E 0 | ⩽ 2 n − 3 , | F | < n − ( ⌊ | E 0 | / 2 ⌋ + 1 ) . If the subgraph induced by E 0 is a linear forest (i.e., pairwise vertex-disjoint paths), then in the graph Q n − F all edges of E 0 lie on a Hamiltonian cycle.