A note on L(2, 1)-labelling of trees

An L(2, 1)-labelling of a graph G is a function from the vertex set V (G) to the set of all nonnegative integers such that {pipe}f(u) - f(v){pipe} ≥ 2 if d _G(u, v) = 1 and {pipe}f(u) - f(v){pipe} ≥ 1 if d _G(u, v) = 2. The L(2, 1)-labelling problem is to find the smallest number, denoted by λ(G), such that there exists an L(2, 1)-labelling function with no label greater than it. In this paper, we study this problem for trees. Our results improve the result of Wang [The L(2, 1)-labelling of trees, Discrete Appl. Math. 154 (2006) 598-603].