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

Let n and k be positive integers with n−k≥2. The arrangement graph An,k is recognized as an attractive interconnection networks. Let x, y, and z be three different vertices of An,k. Let l be any integer with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{A_{n,k}}(\mathbf{x},\mathbf{y}) \le l \le \frac{n!}{(n-k)!}-1-d_{A_{n,k}}(\mathbf{y},\mathbf{z})$\end{document}. 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 dR(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.