Determining the L(2, 1)L(2, 1)-span in polynomial space

A k-L(2, 1)-labeling of a graph is a function from its vertex set into the set {0, . . . , k}, such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices of distance 2 are different. It is known that finding the smallest k admitting the existence of a k-L(2, 1)labeling of any given graph is NP-Complete. In this paper we present an algorithm for this problem, which works in time O((9 + ǫ)n) and polynomial memory, where ǫ is an arbitrarily small positive constant. This is the first exact algorithm for L(2, 1)-labeling problem with time complexity O(cn) for some constant c and polynomial space complexity.