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

• Authors: Wensong Lin;Pu Zhang
• Venue: Discrete Applied Mathematics
• Year: 2012

We initiate research on the multiple distance 2 labeling of graphs in this paper. Let n,j,k be positive 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 2 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 positive 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 2 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. We investigate the basic properties of n-fold L(j,k)-labelings and circular L(j,k)-labelings of graphs. The n-fold circular L(j,k)-labeling numbers of trees, and the hexagonal and p-dimensional square lattices are determined. The upper and lower bounds for the n-fold L(j,k)-labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 both the lower and the upper bounds are sharp. In many cases, the n-fold L(j,k)-labeling numbers of the hexagonal and p-dimensional square lattices are determined. In other cases, upper and lower bounds are provided. In particular, we obtain the exact values of the n-fold L(j,1)-labeling numbers of the hexagonal and p-dimensional square lattices.