On n n -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,k n , j , k be positive integers. An n n -fold L(j,k) L ( j , k ) -labeling  of a graph G G is an assignment f f of sets of nonnegative integers of order n n to the vertices of G G such that, for any two vertices u,v u , 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) u v ? E ( G ) , and |a?b|?k | a ? b | ? k if u u and v v are distance 2 apart. The span of f f is the absolute difference between the maximum and minimum integers used by f f . The n n -fold L(j,k) L ( j , k ) -labeling number of G G is the minimum span over all n n -fold L(j,k) L ( j , k ) -labelings of G G . Let n,j,k n , j , k and m m be positive integers. An n n -fold circular m m -L(j,k) L ( j , k ) -labeling of a graph G G is an assignment f f of subsets of {0,1,,m?1} { 0 , 1 ,  , m ? 1 } of order n n to the vertices of G G such that, for any two vertices u,v u , v and any two integers a?f(u) a ? f ( u ) , b?f(v) b ? f ( v ) , min{|a?b|,m?|a?b|}?j min { | a ? b | , m ? | a ? b | } ? j if uv?E(G) u v ? E ( G ) , and min{|a?b|,m?|a?b|}?k min { | a ? b | , m ? | a ? b | } ? k if u u and v v are distance 2 apart. The minimum m m such that G G has an n n -fold circular m m -L(j,k) L ( j , k ) -labeling is called the n n -fold circular L(j,k) L ( j , k ) -labeling number of G G . We investigate the basic properties of n n -fold L(j,k) L ( j , k ) -labelings and circular L(j,k) L ( j , k ) -labelings of graphs. The n n -fold circular L(j,k) L ( j , k ) -labeling numbers of trees, and the hexagonal and p p -dimensional square lattices are determined. The upper and lower bounds for the n n -fold L(j,k) L ( j , k ) -labeling numbers of trees are obtained. In most cases, these bounds are attainable. In particular, when k=1 k = 1 both the lower and the upper bounds are sharp. In many cases, the n n -fold L(j,k) L ( j , k ) -labeling numbers of the hexagonal and p 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 n -fold L(j,1) L ( j , 1 ) -labeling numbers of the hexagonal and p p -dimensional square lattices.
n n -fold l(j l(j k) l ( j k) l ( j k) l ( j k) l ( j k ) k) l ( j k) l ( j k) l ( j k) l ( j k ) k) l ( j k) l ( j k) l ( j k) l ( j k ) k) l ( j k) l ( j k) l ( j k) l ( j k ) circular l(j n n -fold circular l(j tree k ) -labeling number l(j k ) l(j k ) l ( j k ) -labeling number n n n n n l(j hexagonal lattice p p -dimensional square lattice p p p p p