A sufficient condition for a tree to be (Δ+1)-(2,1)-totally labelable

The -total labeling number of a graph is the width of the smallest range of integers that suffices to label the vertices and the edges of such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least . It is known that every tree with maximum degree has . In this paper, we give a sufficient condition for a tree to have . More precisely, we show that if is a tree with and every -vertex in is adjacent to at most -vertices, then . The result is best possible in the sense that there exist infinitely many trees with and such that each -vertex is adjacent to at most -vertices and at least one -vertex is adjacent to exactly Delta - 2 vertices.