Classification of l(2, 1)-labeling of cartesian products of paths and cycles

The channel assignment problem requires to assign frequency bands to transmitters. An interference may occur if two close transmitters attempt to transmit on close frequencies. In order to avoid such frequency interference, the separation of the channels assigned to them must be sufficient. Moreover, with the distance between two transmitters becoming closer, the difference between two frequency of their channels must be larger apart. Motivated with this application, the concept of L(2, 1)-labeling of graphs was proposed by many researchers. A κ-L(2, 1)-labeling for a graph G is a function f-V(G)→0, 1,..,k such that)f (u).f (v)).2 whenever uv (G) and)f (u).f (v)).1 whenever u and v are at distance two apart. The H-number for G, denoted by (G), is the minimum k such that G admits a κ-L(2, 1)-labeling. In this paper, a computer-aided method based on backtrack search is used to obtain a table of classification of some graphs including Cartesian product of cycles or paths. We conclude that the 6-L(2, 1)-labeling of C7(C7 is unique up to equivalence, and prove that there are exactly two inequivalent 6-L(2, 1)-labelings in Pm(C7form ≥ 3. © 2016 American Scientific Publishers All rights reserved.