Distance Constrained Labeling of Precolored Trees

Graph colorings with distance constraints are motivated by the frequency assignment problem. The so called λ(p,q)-labeling problem asks for coloring the vertices of a given graph with integers from the range {0, 1, ..., λ} so that labels of adjacent vertices differ by at least p and labels of vertices at distance 2 differ by at least q, where p, q are fixed integers and integer λ is part of the input. It is known that this problem is NP-complete for general graphs, even when λ is fixed, i.e., not part of the input, but polynomially solvable for trees for (p,q)=(2,1). It was conjectured that the general case is also polynomial for trees. We consider the precoloring extension version of the problem (i.e., when some vertices of the input tree are already precolored) and show that in this setting the cases q=1 and q > 1 behave di.erently: the problem is polynomial for q=1 and any p, and it is NP-complete for any p > q > 1.