L(d, 1)-labeling is a kind of graph coloring problem from frequency assignment in radio networks, in which adjacent nodes must receive colors that are at least d apart while nodes at distance two from each other must receive different colors. We focus on L(d, 1)-labeling of regular tilings for d≥3 since the cases d=0, 1 or 2 have been researched by Calamoneri and Petreschi. For all three kinds of regular tilings, we give their L (d, 1)-labeling numbers for any integer d≥3. Therefore, combined with the results given by Calamoneri and Petreschi, the L(d, 1)-labeling numbers of regular tilings for any nonnegative integer d may be determined completely.
An L（j, k）-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive integers which are at least j apart, and vertices at distance two receive integers which are at least k apart. Given an L（j, k）-labeling f of G, define the L（j, k） edge span of f, βj,k（G,f） =max{ ｜f（x）-f（y）｜： {x,y}∈E（G）}. The L（j,k） edge span of G, βj,k （G） is min βj,k（ G, f）, where the minimum runs over all L（j, k）-labelings f of G. The real L（.j, k）-labeling of a graph G is a generalization of the L（j, k）-labeling. It is an assignment of nonnegative real numbers to the vertices of G satisfying the same distance one and distance two conditions. The real L（j, k） edge span of a graph G is defined accordingly, and is denoted by βj,k（G）. This paper investigates some properties of the L（j, k） edge span and the real L（j, k） edge span of graphs, and completely determines the edge spans of cycles and complete t-partite graphs.
