The paths embedding of the arrangement graphs with prescribed vertices in given position

Abstract

Let n and k be positive integers with n?k?2. The arrangement graph A n,k is recognized as an attractive interconnection networks. Let x, y, and z be three different vertices of A n,k . Let l be any integer with $d_{A_{n,k}}(\mathbf{x},\mathbf{y}) \le l \le \frac{n!}{(n-k)!}-1-d_{A_{n,k}}(\mathbf{y},\mathbf{z})$ . We shall prove the following existance properties of Hamiltonian path: (1) for n?k?3 or (n,k)=(3,1), there exists a Hamiltonian path R(x,y,z;l) from x to z such that d R(x,y,z;l)(x,y)=l; (2) for n?k=2 and n?5, there exists a Hamiltonian path R(x,y,z;l) except for the case that x, y, and z are adjacent to each other.