  An odd prime labeling is a variation of a prime labeling in which the
vertices of a graph of order~$n$ are labeled with the distinct odd integers $1$
to $2n-1$ so that the labels of adjacent vertices are relatively prime. This
paper investigates many different classes of graphs including disjoint unions
of cycles, stacked prisms, and particular types of caterpillars, by using
various methods to construct odd prime labelings. We also demonstrate progress
toward proving a conjecture that all prime graphs have an odd prime labeling.
