Formula omitted]-Total labelling of trees with sparse vertices of maximum degree

Abstract

The ( 2 , 1 ) -total labelling number λ 2 t ( G ) 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 D Δ ( 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 Δ + 1 ⩽ λ 2 t ( T ) ⩽ Δ + 2 . In this paper, we prove that if 1 ∉ D Δ ( T ) or 2 ∉ D Δ ( T ) , then λ 2 t ( T ) = Δ + 1 . 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 ∉ D Δ ( T ) such that λ 2 t ( T ) = Δ + 2 .