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

Abstract

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