  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.
