L(j,k)-labeling Numbers And Circular Labeling Numbers Of Cartesian Product Of Three Paths

To solve the code assignment problem in computer wireless network,the Cartesian product graph of three paths is used to describe the three-dimensional computer wireless network.By studying the L(j,k)-labeling and circular labeling of the Cartesian product graph of three paths(j<k),the corresponding code assignment strategy of computer wireless network is obtained,so as to alleviate the code shortage problem of computer wireless network.For two vertices u and v in graph G,d(u,v)represents the distance between vertices u and v.Let j,k,m be three nonnegative real numbers,f:V(G)→(0,m)be a mapping,if f satisfies the following conditions:|f(u)-f(v)|≥j if d(u,v)=1;|f(u)-f(v)|≥k if d(u,v)=2,then f is called an m-L(j,k)-labeling of graph G.the minimum span of all L(j,k)-labeling of graph G,denoted by λj,k(G),is called the L(j,k)-labeling number of graph G.Let σ be a nonnegative real number,and circular σ-L(j,k)-labeling of graph G is a mapping,f:V(G)→[0,σ).If f meets the conditions:|f(u)-f(v)|σ≥j if d(u,v)=1;|f(u)-f(v)|σ≥k if d(u,v)=2,where |p-q|σ=min{|p-q|,σ-|p-q|}.f is called a circular σ-L(j,k)-labeling of graph G.The smallest σ of all circular L(j,k)-labels of graph G is called the number of circular L(j,k)-labeling of G,which is represented by σj,k(G).In order to better describe the computer wireless network,this paper studies the threedimensional network graph,and mainly studies the problems of L(j,k)-labeling numbers and circular L(j,k)-labeling numbers of the Cartesian product graph of three paths,where j<k.In this article,by using the symmetry of the Cartesian product graph of three paths,research of subgraph L(j,k)-labeling numbers and circular labeling numbers,combining with the definition of related,and prove the Cartesian product graph of three paths of L(j,k)labeling numbers and circular labeling numbers,where k≥2 j.And study the Cartesian product graph P2□P2□P2 of three paths L(j,k)-circular labeling numbers,where j<k<2j.