Distance three labelings of trees

An L(2,1,1)L(2,1,1)-labeling of a graph GG assigns nonnegative integers to the vertices of GG in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and the aim is to minimize this value. We show that the minimum span of an L(2,1,1)L(2,1,1)-labeling of a tree can be bounded by a lower and an upper bound with difference one. Moreover, we show that deciding whether the minimum span attains the lower bound is an NP-complete problem. This answers a known open problem, which was recently posed by King, Ras, and Zhou as well. We extend some of our results to general graphs and/or to more general distance constraints on the labeling.