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

The $$(2, 1)$$ ( 2 , 1 ) -total labeling number $$\lambda _2^t(G)$$ λ 2 t ( G ) of a graph $$G$$ G is the width of the smallest range of integers that suffices to label the vertices and the edges of $$G$$ G 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 $$2$$ 2 . It is known that every tree $$T$$ T with maximum degree $$\Delta $$ Δ has $$\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2$$ Δ + 1 ≤ λ 2 t ( T ) ≤ Δ + 2 . In this paper, we give a sufficient condition for a tree $$T$$ T to have $$\lambda _2^t(T) = \Delta + 1$$ λ 2 t ( T ) = Δ + 1 . More precisely, we show that if $$T$$ T is a tree with $$\Delta \ge 4$$ Δ ≥ 4 and every $$\Delta $$ Δ -vertex in $$T$$ T is adjacent to at most $$\Delta - 3$$ Δ - 3 $$\Delta $$ Δ -vertices, then $$\lambda _2^t(T) = \Delta + 1$$ λ 2 t ( T ) = Δ + 1 . The result is best possible in the sense that there exist infinitely many trees $$T$$ T with $$\Delta \ge 4$$ Δ ≥ 4 and $$\lambda _2^t(T) = \Delta + 2$$ λ 2 t ( T ) = Δ + 2 such that each $$\Delta $$ Δ -vertex is adjacent to at most $$\Delta -2$$ Δ - 2 $$\Delta $$ Δ -vertices and at least one $$\Delta $$ Δ -vertex is adjacent to exactly $$\Delta -2$$ Δ - 2 vertices.