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.

所述ķ的顺序-coprime图表Ñ是与顶点集合的曲线\（\ {K，K + 1，\ ldots，K + N-1 \} \） ，其中两个顶点相邻当且仅当它们彼此互质。我们刻画哈密顿k互质图。作为一个特殊情况，Tout 等人的两个猜想。(Natl Acad Sci Lett 11:365–368, 1982) 和 Schroeder (Graph Comb 38:119–140, 2019) 关于 2-正则图的素数标记如下。具有n个顶点的图的素数标记是用与\(\{1, 2,\ldots , n\}\)不同的整数标记其顶点，使得任何两个相邻顶点的标签相对素数.