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

Abstract

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