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

Abstract

The (2, 1)-total labeling number λ t 2 (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 has + 1 ≤ λ t 2 (T ) ≤ + 2. In this paper, we give a sufficient condition for a tree T to have λ t 2 (T ) = + 1. More precisely, we show that if T is a tree with ≥ 4 and every -vertex in T is adjacent to at most − 3 -vertices, then λ t 2 (T ) = + 1. The result is best possible in the sense that there exist infinitely many trees T with ≥ 4 and λ t 2 (T ) = + 2 such that each -vertex is adjacent to at most − 2 -vertices and at least one -vertex is adjacent to exactly − 2 vertices.