Long cycles and paths in distance graphs

Abstract

For $n\in \mathbb{N}$ and $D\subseteq \mathbb{N}$, the distance graph $P_n^D$ has vertex set $\{ 0,1,\ldots,n-1\}$ and edge set $\{ ij\mid 0\leq i,j\leq n-1, |j-i|\in D\}$. 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 $c_D$ such that the greatest common divisor of the integers in $D$ is $1$ if and only if for every $n$, $P_n^D$ has a component of order at least $n-c_D$ if and only if for every $n$, $P_n^D$ has a cycle of order at least $n-c_D$. Furthermore, we discuss some consequences and variants of this result