Real Number Graph Labeling With A Condition At Distance Two

The channel assignment problem,originates from the radio channel in-terference,is to assign channels to the transmitters in a network in a waywhich avoids interference and uses the spectrum as efficiently as possible．Ra-dio channel assignments with multiple levels of interference can be modeledusing graph theory．Given a graph G with diameter at least p and real numbers k₁,k₂,…,k_p≥0,a L(k₁,k₂,…,k_p)-labeling of G assigns a real number f(x)≥0 to eachvertex x such that the labels of any two different vertices u and v differ byat least k_i,where i=d_G(u,v)≤p is the distance between u and v．Onegoal of L(k₁,k₂,...,k_p)-labelings is to determine the infimum span,denotebyλ(Gï¼›k₁,k₂,...,k_p),over all such labelings f．In this paper,we study the value ofλ(Gï¼›k,1)with k≤1/2 for triangle-free graph．The main results are as follows:(1)．If a graph G is triangle-free,then(2)．If G is bipartite,then(3)．If T is a tree,thenFurthermore．a class of trees which meetâ–³-1 is determined．...