Long cycles and paths in distance graphs

For n ∈ N and D ⊆ N, the distance graph P n has vertex set {0, 1, . . . , n− 1} and edge set {ij | 0 ≤ i, j ≤ n − 1, |j − i| ∈ D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs. A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a set D, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, P n has a component of order at least n−cD if and only if for every n, P n has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result.