Multiple L(j,1)-labeling of the triangular lattice

Let n,j,k be nonnegative integers. An n-foldL(j,k)-labeling of a graph G is an assignment f of sets of nonnegative integers of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), |a−b|≥j if uv∈E(G), and |a−b|≥k if u and v are distance two apart. The span of f is the absolute difference between the maximum and minimum integers used by f. The n-fold L(j,k)-labeling number of G is the minimum span over all n-fold L(j,k)-labelings of G.Let n,j,k and m be nonnegative integers. An n-fold circular m-L(j,k)-labeling of a graph G is an assignment f of subsets of {0,1,…,m−1} of order n to the vertices of G such that, for any two vertices u,v and any two integers a∈f(u), b∈f(v), min{|a−b|,m−|a−b|}≥j if uv∈E(G), and min{|a−b|,m−|a−b|}≥k if u and v are distance two apart. The minimum m such that G has an n-fold circular m-L(j,k)-labeling is called the n-fold circular L(j,k)-labeling number of G.This paper provides upper and lower bounds for the n-fold L(j,1)-labeling number and the n-fold circular L(j,1)-labeling number of the triangular lattice and determines the n-fold L(2,1)-labeling number and n-fold circular L(2,1)-labeling number of the triangular lattice for n≥3.