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

Abstract

The $$(2, 1)$$(2,1)-total labeling number $$\lambda _2^t(G)$$?2t(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≤?2t(T)≤Δ+2. In this paper, we give a sufficient condition for a tree $$T$$T to have $$\lambda _2^t(T) = \Delta + 1$$?2t(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 $$\lambda _2^t(T) = \Delta + 1$$?2t(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 $$\lambda _2^t(T) = \Delta + 2$$?2t(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.