A new sufficient condition for a tree T to have the (2, 1)-total number $$\Delta +1$$ Δ + 1

Abstract

A k-(2, 1)-total labelling of a graph G is a mapping $$f: V(G)\cup E(G)\rightarrow \{0,1,\ldots ,k\}$$f:V(G)źE(G)ź{0,1,ź,k} such that adjacent vertices or adjacent edges receive distinct labels, and a vertex and its incident edges receive labels that differ in absolute value by at least 2. The (2, 1)-total number, denoted $$\lambda _2^t(G)$$ź2t(G), is the minimum k such that G has a k-(2, 1)-total labelling. Let T be a tree with maximum degree $$\Delta \ge 7$$Δź7. A vertex $$v\in V(T)$$vźV(T) is called major if $$d(v)=\Delta $$d(v)=Δ, minor if $$d(v)<\Delta $$d(v)<Δ, and saturated if v is major and is adjacent to exactly $$\Delta - 2$$Δ-2 major vertices. It is known that $$\Delta + 1 \le \lambda _2^t(T)\le \Delta + 2$$Δ+1≤ź2t(T)≤Δ+2. In this paper, we prove that if every major vertex is adjacent to at most $$\Delta -2$$Δ-2 major vertices, and every minor vertex is adjacent to at most three saturated vertices, then $$\lambda _2^t(T) = \Delta + 1$$ź2t(T)=Δ+1. The result is best possible with respect to these required conditions.