Hamiltonicity of a Coprime Graph

Abstract

The k-coprime graph of order n is the graph with vertex set \(\{k, k+1, \ldots , k+n-1\}\) in which two vertices are adjacent if and only if they are coprime. We characterize Hamiltonian k-coprime graphs. As a particular case, two conjectures by Tout et al. (Natl Acad Sci Lett 11:365–368, 1982) and by Schroeder (Graph Comb 38:119–140, 2019) on prime labeling of 2-regular graphs follow. A prime labeling of a graph with n vertices is a labeling of its vertices with distinct integers from \(\{1, 2,\ldots , n\}\) in such a way that the labels of any two adjacent vertices are relatively prime.