An O(n1.75) algorithm for L(2,1)-labeling of trees

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x)−f(y)|≥2 if x and y are adjacent and |f(x)−f(y)|≥1 if x and y are at distance 2 for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,…,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible L(2,1)-labelings. It is known that this problem is NP-hard even for graphs of treewidth 2. Tree is one of a few classes for which the problem is polynomially solvable, but still only an O(Δ4.5n) time algorithm for a tree T has been known so far, where Δ is the maximum degree of T and n=|V(T)|. In this paper, we first show that an existent necessary condition for λ(T)=Δ+1 is also sufficient for a tree T with , which leads to a linear time algorithm for computing λ(T) under this condition. We then show that λ(T) can be computed in O(Δ1.5n) time for any tree T. Combining these, we finally obtain an O(n1.75) time algorithm, which substantially improves upon previously known results.