EQUITABLE EDGE COLORING ON TENSOR PRODUCT OF GRAPHS

Abstract. A graph G is edge colored if different colors are assigned to its edges or lines, in the order of neighboring edges are allotted with least diverse k-colors. If each of k-colors can be partitioned into color sets and differs by utmost one, then it is equitable. The minimum of k-colors required is known as equitably edge chromatic number and symbolized by χ=(G). Further the impression of equitable edge coloring was first initiated by Hilton and de Werra in 1994. In this paper, we ascertain the equitable edge chromatic number of Pm ⊗ Pn, Pm ⊗ Cn and K1,m ⊗K1,n.


Introduction
In the midst of various coloring concepts of graphs, the motive of equitability in edge coloring on tensor product of graphs is an inventive approach. Graphs considered in this paper are of simple …nite sets V and E. Each element of V is called its vertices and the elements of E are called its edges, which are the unordered pair of vertices. Therefore G(V; E) is a graph. We use the standard notation of graph theory [1,2]. The minimum number of colors needed to color edges of a graph G is utmost its maximum degree. Since all edges incident to the same vertex must be alloted with distinct colors. Noticeably 0 (G) (G). In 1964, Vizing [3] conjectured that for every simple graph 0 (G) (G) + 1. In 1973, Meyer [4] presented the concept of equitable vertex coloring and its equitable EQUITABLE EDGE COLO RING ON TENSOR PRODUCT OF GRAPHS 1337 chromatic number, which opened the way for introducing equitability in the …elds of edge and total coloring.
The concept of equitable edge coloring was de…ned by Hilton and de Werra [5] and the tensor product of graph was de…ned by P.M.Weichsel [6]. We have merged both these conception and resolute the equitable edge chromatic number of P m P n , P m C n and K 1;m K 1;n . The combined component of each of these graphs enlarges as a new structured graph and has wider applications in the areas of networks, scheduling and assignment domains.

Preliminaries
De…nition 2.1. An edge coloring of a graph G is a function f : E(G) ! C, where C is a set of distinct colors. For any positive integer k, a k-edge coloring is an edge coloring that uses exactly k di¤ erent colors. A proper edge coloring of a graph is an edge coloring such that no two adjacent edges are assigned the same color. Thus a proper edge coloring f of G is a function f : E(G) ! C such that f (e) 6 = f (e 0 ) whenever edges e and e 0 are adjacent in G.
De…nition 2.2. The chromatic index of a graph G, denoted 0 (G), is the minimum number of di¤ erent colors required for a proper edge coloring of G. The graph G is k-edge-chromatic if 0 (G) = k.     p; p 1 ( mod 2) ; p 1; p 0 ( mod 2) ; Lemma 2.9. [7] Let G(V; E) be a connected graph. If there are two adjacent vertices with maximum degree, then 0 as (G) (G) + 1.
Proof. Let V (P m ) = fu i : 1 i mg and V (P n ) = fv j : 1 j ng. By the de…niton of tensor product, V (P m P n ) = fu i v j : 1 i m; 1 j ng and Let e (i)(j);(k)(l) be the edge of P m P n connecting the vertices u i v j and u k v l of P m P n . Therefore e (i)(j);(k)(l) 2 E (P m P n ) if and only if jk ij = jl ij = 1. Since P m P n is isomorphic to P n P m . Without loss of generality, we assume m n for all cases of m and n. Now let us partition E (P m P n ) for the following cases. Case (i) Both m and n are odd In this partition jE 1 j = jE 2 j = jE 3 j = jE 4 j = 2 m 1 2 n 1 2 and satis…es jjE i j jE j jj 1 for i 6 = j. Case (ii) When m is odd and n is even which infers jjE i j jE j jj 1 for i 6 = j. Case (iii) When m is even and n is odd which signi…es jjE i j jE j jj 1 for In all the cases by observing the su¢ xes of the edges of E i and E j (i 6 = j), it is inferred that there is no common edges in E i and E j (i 6 = j). i.e, E i \ E j = for i 6 = j. Clearly E i 's are pair wise mutually disjoint, also Here P m P n is equitably edge colorable with 4 colors. Hence 0 = (P m P n ) 4. Since = 4, we have 0 = (P m P n ) 0 (P m P n ) = 4. This implies 0 = (P m P n ) 4. Therefore 0 = (P m P n ) = 4.
Let e (i)(j);(k)(l) be the edge of P m C n connecting the vertices u i v j and u k v l of P m C n .
Therefore e (i)(j);(k)(l) 2 E (P m C n ) if and only if jk ij = jl ij = 1. Since P m C n is isomorphic to C n P m . Without loss of generality, we assume m n for all cases of m and n. Now let us partition E (P m C n ) for the following cases. Case (i) Both m and n are odd . and deduce that jjE i j jE j jj 1 for i 6 = j. Case (ii) When m is odd and n is even . and assures thatjjE i j jE j jj 1 for i 6 = j. Case (iii) When m is even and n is odd