The L(2, 1)-labeling on ࢻ-product of Graphs

The L(2, 1)-labeling (or distance two labeling) of a graph G is an integer labeling of G in which two vertices at distance one from each other must have labels differing by at least 2 and those at distance two must differ by at least 1. The L(2, 1)-labeling numberλሺGሻ of G is the smallest number k such that G has an L(2, 1)-labeling max{ (): ()} f v v V G k ∈ = with max ሺfሺvሻ: v ‫א‬ VሺGሻ ൌ k. In this paper, upper bound for the L(2, 1)-labeling number for the α-product of two graphs has been obtained in terms of the maximum degrees of the graphs involved. Degrees of vertices, vertex of maximum degree and number of vertices of maximum degree have been discussed in the α-product of two graphs. 1. Introduction The concept of L(2, 1)-labeling in graph come into existence with the solution of frequency assignment problem. In fact, in this problem a frequency in the form of a non-negative integer is to assign to each radio or TV transmitters located at various places such that communication does not interfere. Hale [6] was first person who formulated this problem as a graph vertex coloring problem.


Introduction
The concept of L(2, 1)-labeling in graph come into existence with the solution of frequency assignment problem. In fact, in this problem a frequency in the form of a nonnegative integer is to assign to each radio or TV transmitters located at various places such that communication does not interfere. Hale [6] was first person who formulated this problem as a graph vertex coloring problem.
Griggs and Yeh [5] provided an upper bound 2 2 ∆ + ∆ for a general graph with the maximum degree ∆ . Later, Chang and Kuo [1] improved the upper bound to 2 ∆ + ∆ , while Kral and Skrekarski [10] reduced the upper bound to 2 1 ∆ + ∆ − . Furthermore, recently Gonccalves [4] proved the bound 2 2 ∆ + ∆ − which is the present best record. If G is a diameter 2 graph, then 2 ( ) G λ ≤ ∆ . The upper bound is attainable by Moore graphs (diameter 2 graphs with order 2 1 ∆ + ). (Such graphs exist only if 2,3,7 ∆ = and possibly 57; [5]). Thus Griggs and Yeh [5] conjectured that the best bound is 2 ∆ for any graph G with the maximum degree 2 ∆ ≥ . (This is not true for 1 ∆ = . For example, Graph products play an important role in connecting many useful networks. Klavzar and Spacepan [9] have shown that 2 ∆ -conjecture holds for graphs that are direct or strong products of nontrivial graphs. After that Shao, et al. [13] have improved bounds on the L(2, 1)-labeling number of direct and strong product of nontrivial graphs with refined approaches. Shao and Shang [15] also consider the graph formed by the Cartesian sum of graphs and prove that the λ -number of L(2, 1)-labeling of this graph satisfies the 2 ∆ -conjecture (with minor exceptions).
In this paper, we have considered the graph formed by the α -product of graphs [3] and obtained a general upper bound for L(2, 1)-labeling number in term of maximum degree of the graphs. In the case of α -product of graphs, L(2, 1)-labeling number of graph holds Griggs and Yeh's conjecture [5] with minor exceptions.

A labeling algorithm
A subset X of ( ) V G is called an i -stable set (or i -independent set) if the distance between any two vertices in X is greater than i , i.e. { ( , ) , , 1-stable set is a usual independent set. A maximal 2-stable subset X of a set Y is a 2-stable subset of Y such that X is not a proper subset of any 2-stable subset ofY .
Chang and Kuo [1] proposed the following algorithm to obtain an L(2, 1)-labeling and the maximum value of that labeling on a given graph.

Algorithm:
Output: The value k is the maximum label. Idea: In each step i , find a 2-maximal 2-stable set from the unlabeled vertices that are distance at least two away from those vertices labelled in the previous step. Then label all the vertices in that 2-stable with i in current stage. The label i starts from 0 and then increases by 1 in each step. The maximum label k is the final value of i . Iteration: The L(2, 1)-Labeling on ߙ-Product of Graphs 31 1. Determine i Y and i X .
2. Label the vertices of i Let u be a vertex with largest label k obtained by above Algorithm. We have the following sets on the basis of Algorithm just defined above.
I is the set of labels of the neighbourhood of the vertex u .
I is the set of labels of the vertices at distance at most 2 from the vertex u .
i.e. 3 I consists of the labels not used by the vertices at distance at most 2 from the vertex u .
Then Chang and Kuo showed that In order to find k , it suffices to estimate 1 2 B I I = + in terms of ( ) G ∆ . We will investigate the value B with respect to a particular graph (α -product of two graphs). The notations which have been introduced in this section will also be used in the following sections.

The ࢻ -product of graphs
For example, we consider the Figure 1. Now, we state and prove the following corollary to find out the degree of any vertex of α -product of two graphs.
From the above two cases, it can be written as  u v and ( , ) u v . Hence for any vertex in G at distance 2 from ( , ) altogether. By the above analysis, the number By the analysis of the above two cases, the best possible for the number of vertices with distance 2 from Hence for the vertex u α , the number of vertices with distance 1 from u α is no greater than ∆ . The number of vertices with distance 2 from u α is no greater than