Panpositionable hamiltonicity and panconnectivity of the arrangement graphs

The arrangement graph An,k is a generalization of the star graph. It is more flexible in its size than the star graph. There are some results concerning hamiltonicity and pancyclicity of the arrangement graphs. In this paper, we propose a new concept called panpositionable hamiltonicity. A hamiltonian graph G is panpositionable if for any two different vertices x and y of G and for any integer l   satisfying d(x,y)⩽l⩽|V(G)|-d(x,y)d(x,y)⩽l⩽|V(G)|-d(x,y), there exists a hamiltonian cycle C of G such that the relative distance between x and y on C is l. A graph G is panconnected if there exists a path of length l joining any two different vertices x and y   with d(x,y)⩽l⩽|V(G)|-1d(x,y)⩽l⩽|V(G)|-1. We show that An,k is panpositionable hamiltonian and panconnected if k ⩾ 1 and n − k ⩾ 2.