On nn-fold L(j,k)L(j,k)-and circular L(j,k)L(j,k)-labelings of graphs

We initiate research on the multiple distance 2 labeling of graphs in this paper.Let n,j,kn,j,k be positive integers. An nn-fold  L(j,k)L(j,k)-labeling   of a graph GG is an assignment ff of sets of nonnegative integers of order nn to the vertices of GG such that, for any two vertices u,vu,v and any two integers a∈f(u)a∈f(u), b∈f(v)b∈f(v), |a−b|≥j|a−b|≥j if uv∈E(G)uv∈E(G), and |a−b|≥k|a−b|≥k if uu and vv are distance 2 apart. The span of ff is the absolute difference between the maximum and minimum integers used by ff. The nn-fold L(j,k)L(j,k)-labeling number of GG is the minimum span over all nn-fold L(j,k)L(j,k)-labelings of GG.Let n,j,kn,j,k and mm be positive integers. An nn-fold circular mm-L(j,k)L(j,k)-labeling of a graph GG is an assignment ff of subsets of {0,1,…,m−1}{0,1,…,m−1} of order nn to the vertices of GG such that, for any two vertices u,vu,v and any two integers a∈f(u)a∈f(u), b∈f(v)b∈f(v), min{|a−b|,m−|a−b|}≥jmin{|a−b|,m−|a−b|}≥j if uv∈E(G)uv∈E(G), and min{|a−b|,m−|a−b|}≥kmin{|a−b|,m−|a−b|}≥k if uu and vv are distance 2 apart. The minimum mm such that GG has an nn-fold circular mm-L(j,k)L(j,k)-labeling is called the nn-fold circular L(j,k)L(j,k)-labeling number of GG.We investigate the basic properties of nn-fold L(j,k)L(j,k)-labelings and circular L(j,k)L(j,k)-labelings of graphs. The nn-fold circular L(j,k)L(j,k)-labeling numbers of trees, and the hexagonal and pp-dimensional square lattices are determined. The upper and lower bounds for the nn-fold L(j,k)L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1k=1 both the lower and the upper bounds are sharp. In many cases, the nn-fold L(j,k)L(j,k)-labeling numbers of the hexagonal and pp-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the nn-fold L(j,1)L(j,1)-labeling numbers of the hexagonal and pp-dimensional square lattices.