On Hamiltonian paths in distance graphs

For a finite set D⊆ND⊆N with gcd(D)=1, we prove the existence of some n∈Nn∈N such that the distance graph PnD with vertex set {0,1,…,n−1}{0,1,…,n−1} in which two vertices uu and vv are adjacent exactly if |u−v|∈D|u−v|∈D, has a Hamiltonian path with endvertices 00 and n−1n−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].