Prime labelings of infinite graphs

Abstract

A finite graph on $n$ vertices has a prime labeling provided there is a way to label the vertices with the integers 1 through $n$ such that every pair of adjacent vertices has relatively prime labels. We extend the definition of prime labeling to infinite graphs and give a simple necessary and sufficient condition for an infinite graph to have a prime labeling. We then measure the complexity of prime labelings of infinite graphs using techniques from computability theory to verify that our condition is as simple as possible.