Pair Labelings of Graphs
David R. Guichard, John W. Krussel · SIAM Journal on Discrete Mathematics · 1992
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 $\{ 0,1, \cdots ,n - 1 \}$ 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.