Coprime and prime labelings of graphs

A coprime labeling of a simple graph of order \(n\) is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first \(n\) positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss coprime labelings for complete bipartite graphs.