On Hamiltonian paths in distance graphs

For a finite set D ⊆ N with gcd ( D ) = 1 , we prove the existence of some n ∈ N such that the distance graph P n D with vertex set { 0 , 1 , … , n − 1 } in which two vertices u and v are adjacent exactly if | u − v | ∈ D , has a Hamiltonian path with endvertices 0 and n − 1 . This settles a conjecture posed by Penso et al. [L.D. Penso, D. Rautenbach, J.L. Szwarcfiter, Long cycles and paths in distance graphs, Discrete Math. 310 (2010) 3417–3420].