On L(j,k)-labelings And L(j,k,l)-labelings Of Trees

Let G=(V(G),E(G)) be a graph. For two vertices u, v∈V(G), we denote d(u,v) the distance between u and v. Let j, k be positive integers with j≥k. An L(j,k)-labeling for G is a function f:V(G)→{0,1,2,...}, such that for any two vertices u and v,|f(u)-f(v)|is at least j if d(u, v)=1; and is at least k if d(u,v)=2. The span of is the difference between the largest and the smallest numbers in f(V). The λj,k-number for G, denoted by λj,k(G), is the minimum span over all L(j,k)-labellings of G. As the definition of L(j,k)-labeling, we can also define L(j, k,l)-labeling of graph G. Let j, k, l be positive integers with j≥k≥l. An L(j,k,l)-labeling for G is a function f:V(G)→{0,1,2,...}, such that for any two vertices u and v,|f(u)-f(v)|is at least j if d{u, v)=1; and is at least k if d{u, v)=2; and is at least l if d(u,v)=3. The L(j,k)-labeling problem of tree, in particular the L(2,1) case, has been extensively studied. But when it comes to L(j, k,l)-labeling problem, we know nothing but few conclusions. In this paper we verify the conjecture brought by J.Georges and D.W.Mauro in1995in their paper[17] is correct for Caterpillar. We also investigate L(j,2,1)-labeling problem of simple graph such as path Pn and circle Cn,n>4. At last we find out the bound of λj,2,1(T) for tree T with maximum degree△(△>3).