Summary: Given a graph \(G\) and positive integer \(d\), the pair-labeling number \(r^*(G,d)\) is the minimum \(n\) such that each vertex in \(G\) can be assigned a pair of numbers from \(\{0,1,\dots,n-1\}\) so that any two numbers used at adjacent vertices differ by at least \(d\) modulo \(n\). All possible values of \(r^*(G,d)\), given the chromatic number of \(G\), are determined.