ST-COLORING OF SOME PRODUCTS OF GRAPHS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. For a finite set T of non negative integers containing zero, a function c : V (G)→ Z+ ⋃ {0} is said to be a ST -coloring of the graph G = (V,E), if | c(x)− c(y) | is not in T for any any edge (x,y) and for any two distinct edges (x,y) and (u,v), | c(x)−c(y) |6=| c(u)−c(v) |. spST (G) is the minimum of the difference between the largest and smallest colors assigned over all the vertices and espST (G) is the minimum of the maximum difference between the colors assigned to the vertices of an edge over all the edges of the graph, where the minimum is taken over all ST -coloring c. Here we establish some results related to ST -chromatic number, span and edge span of some graph products namely, Tensor product, Cartesian product and Corona product of graphs.


INTRODUCTION
Graph coloring is one of the most important and extensively well known researched area in the field of graph theory. It is an important subfield of graph theory having various applications.
Assignment of frequencies to various channels is one of such famous and well known problems in the field of telecommunication. Channel assignment problem [7] can be modelled with the help of graph, where transmitter channels will be considered as some vertices and if there is any interference between any two transmitters, then that interference can be considered as an edge.
T -coloring is one kind of vertex coloring, which was introduced by W. K. Hale [3] by setting up an interrelation between graph coloring and the channel assignment problem. For a fixed set T of non negative integers containing zero, T -coloring of a graph G = (V, E), is a function f : V (G) → Z + ∪ {0}, such that the absolute values of the differences of the colors or non negative integers assigned to any two distinct vertices must not be in the fixed set T . T -chromatic number, T -span and T -edge span are some important measures of a T -coloring. T -chromatic number is the minimum number of colors or non negative integers required for an efficient Tcoloring or the order of the T -coloring f and T -span is the maximum absolute differences of the non negative integers or colors assigned to any two distinct vertices. Whereas, T −edge span is the maximum absolute differences of the non-negative integers or colors assigned to two vertices of all the edges. For more about T -colorings, we refer to [1,2,5,7,9,[11][12][13]. A particular There are various types of graph products, such as, Cartesian product, Tensor product also called as direct product, Lexicographic product also called as graph composition, Strong product, etc. For more details on products of graph, see [4]. All graphs considered in this paper are finite, simple and undirected.
In this paper, we consider strong T -colorings on some graph products, viz. Tensor product, Cartesian product and Corona product of graphs. We start with some preliminary results in the next section, followed by the main results of ST -coloring of Tensor product, Cartesian product and Corona product in section 3. In section 4, the conclusion of the paper is drawn.  Observation 1:

PRELIMINARIES
For more results of ST -colorings, we refer to [6,10].

MAIN RESULTS
3.1. ST-coloring of Tensor Product of Graphs.
Theorem 3.1. For all finite set T of positive integers containing zero and for any two graphs (ii) Let f and g are two ST -colorings of the graphs G 1 and G 2 respectively, such that sp therefore , we have, From equations (2) and (3) (iii) The proof can be obtained by using the definition of ST − edge span and proceeding in the similar to the proof of (ii).
Corollary 3.1. For any T -set, If G 1 and G 2 are two subgraphs of their Tensor products, Proof. (i) If G 1 and G 2 are subgraphs of (G 1 × G 2 ), then by using theorem 2.1 (i) Hence, by using theorem 3.1 (ii) and equation (5), (ii) If G 1 and G 2 are subgraphs of (G 1 × G 2 ), then by using theorem 2.1 (ii) Hence, by using theorem 3.1 (iii) and equation (6),

ST-coloring of Cartesian Product of Graphs.
Theorem 3.2. For any T -sets of positive integers containing zero and for any two graphs G 1 Proof. (i) Since the Cartesian product of two graphs G 1 and G 2 , G 1 G 2 contains subgraphs that are isomorphic to both G 1 and G 2 . Hence, (ii) Since the Cartesian product of two graphs G 1 and G 2 , G 1 G 2 contains subgraphs that are isomorphic to both G 1 and G 2 . Hence, by using theorem 2.
Let f and g are two ST -colorings of G 1 and G 2 respectively such that sp f ST (G 1 ) = sp ST (G 1 ) and sp g ST (G 2 ) = sp ST (G 2 ). Let, c be a coloring on (G 1 G 2 ) defined as . Then we have two cases: either u 1 = u 2 and v 1 is adjacent to v 2 in G 2 or v 1 = v 2 and u 1 is adjacent to u 2 in G 1 .
Case I: Hence, Case II: v 1 = v 2 and u 1 is adjacent to Hence, Hence, from equations (9) and (10) Thus, from equations (8) and (11) (iii) The proof can be obtained by using the definition of ST − edge span and proceeding in the similar to the proof of (ii).
Corollary 3.2. If G 1 and G 2 are two graphs such that G 1 is a subgraph of G 2 , then Proof. (i) Since, G 1 is a subgraph of G 2 , then,by using theorem 2.1 (i), sp ST (G 1 ) ≤ sp ST (G 2 ). Thus, (ii) Since, G 1 is a subgraph of G 2 , then, by using theorem 2. Thus, Now the following is the generalization of theorem 3.2 (ii) and 3.2 (iii) to Cartesian product of n graphs G 1 , G 2 , G 3 , ..., G n as follows: Proof. (i) For any T-sets of positive integers containing zero, We shall prove this by the method of induction. Let, G 1 and G 2 are two graphs, then by using theorem 3.
Let, the result is true for any k ≤ n − 1 such graphs. Then, (ii) For any T-sets of positive integers containing zero, We shall prove this by the method of induction. Let, G 1 and G 2 are two graphs, then by using theorem 3.2 (iii),

ST-chromatic number of Corona Product of Graphs.
Theorem 3.4. For any two vertex disjoint graph G 1 and G 2 and for any T -set, Proof. Let T be a set of positive integers containing zero. Let, Let, c be a coloring defined on G 1 • G 2 as, Now, we need to show that If (v i j , v lm ), (v ab , v cd ) are adjacent then equation (13) holds. Hence assume that, (v i j , v lm ), (v ab , v cd ) are non-adjacent edges. Hence, i + j, l + m, a + b, c + d are distinct positive integers. Without loss of generality, let i + j is the largest and c + d is the smallest integer. Then, If possible equation (13) is not true. Then, The proof for c Hence, c is a ST-coloring. In G 1 • G 2 , each of u i j 's of G 1 are adjacent to all v i j 's of i th copy of G 2 . Then for all j = 1 to n, ⇒ all the vertices of i th copies of G 2 in G 1 • G 2 will have distinct colors. Hence | v(G 2 ) |= n no's of colors will be required to color the vertices of i th copy of G 2 in G 1 • G 2 for ST-coloring and χ ST (G 1 ) is the minimum number of colors for ST -coloring of G 1 . Hence, ⇒ all the vertices of i th copies of G in K m • G will have distinct positive integers or colors.
Hence | v(G) |= n no's of positive integers or colors will be required to color the vertices of i th

CONCLUSION
In the paper, we establish some results related to ST -chromatic number of some graph products, viz. Tensor products, Cartesian products and Corona products. We also find few results