On the Computational Complexity of the L(2, 1)-Labeling Problem for Regular Graphs

An L(2,1)-labeling of a graph of span t is an assignment of integer labels from {0,1,...,t} to its vertices such that the labels of adjacent vertices differ by at least two, while vertices at distance two are assigned distinct labels. We show that for all k ≥ 3, the decision problem whether a k-regular graph admits an L(2,1)-labeling of span k+2 is NP-complete. This answers an open problem of R. Laskar.