Total edge irregularity strength of trees

A total edge-irregular k-labelling : V (G) E(G) {1, 2, . . . , k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f ) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree on p vertices tes(T ) = max{ (p + 1)/3 , ( + 1)/2 }.