(2,1)(2,1)-total labeling of trees with large maximum degree

A kk-(2, 1)-total labeling of a graph GG is to label the vertices and the edges of GG using integers from 0 to kk such that all adjacent vertices as well as edges receive different labels, and the difference between the labels of a vertex and its incident edges is at least 2. The (2,1)(2,1)-total labeling number λ2t(G) is the smallest integer kk such that GG has a kk-(2, 1)-total labeling. It is known that λ2t(T), where TT is a tree with maximum degree ΔΔ, equals to either Δ+1Δ+1 or Δ+2Δ+2. In this paper, we provide a sufficient condition for a tree TT to have λ2t(T)=Δ+1 when Δ≥9Δ≥9.