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

Haina Sun,D Mingxing,X Guosheng

doi:10.1051/matecconf/20153117006

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 34 ( ) DT , then T is Type 1.

Highlights

Motivated by the Frequency Channel Assignment problem, Griggs and Yeh[1]introduced the L(2 1) -labelling of graphs
This notion was subsequently generalized to the L( p q) -labelling problem of graphs
An L( p q) labelling of a graph G is a function f from its vertex set V (G) to the set {0 1 k} such that _ f (x) f ( y) _t p if x and y are adjacent, and _ f (x) f ( y) _t q if x and y are at distance 2

Summary

Introduction

Motivated by the Frequency Channel Assignment problem, Griggs and Yeh[1]introduced the L(2 1) -labelling of graphs. An L( p q) labelling of a graph G is a function f from its vertex set V (G) to the set {0 1 k} such that _ f (x) f ( y) _t p if x and y are adjacent, and _ f (x) f ( y) _t q if x and y are at distance 2. Let '(G) (or ' ) denote the maximum degree of a graph G . L(2 1) total labelling number of a generalized star is

Results

Conclusion