An O (n^1.75) 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 (Δ^4.5 n) 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 Δ = Ω (sqrt(n)), 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 (Δ^1.5 n) time for any tree T. Combining these, we finally obtain an O (n^1.75) time algorithm, which substantially improves upon previously known results.