L(j,k)-labeling numbers of square of paths

Abstract

For j≤k, the L(j,k)-labeling arose from code assignment problem. That is, let j, k and m be positive numbers, an m-L(j,k)-labeling of a graph G is a mapping f:V(G)→[0,m] such that |f(u)−f(v)|≥j if d(u,v)=1, and |f(u)−f(v)|≥k if d(u,v)=2. The span of f is the difference between the maximum and the minimum numbers assigned by f. The L(j,k)-labeling number of G, denoted by λj,k(G), is the minimum span over all L(j,k)-labelings of G. The kth power Gk of an undirected graph G is the graph with the vertex set of G in which two vertices are adjacent when their distance in G is at most k. In this paper, the L(j,k)-labeling numbers of Pn2 are determined for j≤k.