The research of (2,1)-total labelling of trees basen on Frequency Channel Assignment problem

Abstract

Abstract. Let T be a tree, Let DΔ (T) denote the set of integers k for which there exist two distinct vertices of maximum degree of distance at k in T. The (2,1)-total labelling number 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. In this paper, we prove that if T is a tree with Δ≥5 and 3 4 ( ) D T ' , then T is Type 1.