(2,1)-Total labelling of trees with sparse vertices of maximum degree

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices, or two adjacent edges, have the same label and the difference between the labels of a vertex and its incident edges is at least 2.Let T be a tree with maximum degree Δ⩾4. Let DΔ(T) denote the set of integers k for which there exist two distinct vertices of maximum degree of distance at k in T. It was known that . In this paper, we prove that if 1∉DΔ(T) or 2∉DΔ(T), then . The result is best possible in the sense that, for any fixed integer k⩾3, there exist infinitely many trees T with Δ⩾4 and k∉DΔ(T) such that .