Equitable Coloring on Total Graph of Bigraphs and Central Graph of Cycles and Paths

1 Department of Mathematics, University College of Engineering Nagercoil, Anna University of Technology Tirunelveli (Nagercoil Campus), Nagercoil 629 004, Tamil Nadu, India 2 Department of Mathematics, R.V.S College of Engineering and Technology, Coimbatore 641 402, Tamil Nadu, India 3 Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore 641 062, Tamil Nadu, India


Introduction
The central graph 1, 2 C G of a graph G is formed by adding an extra vertex on each edge of G, and then joining each pair of vertices of the original graph which were previously nonadjacent.
The total graph 3, 4 of G has vertex set V G ∪ E G and edges joining all elements of this vertex set which are adjacent or incident in G.
If the set of vertices of a graph G can be partitioned into k classes V 1 , V 2 , . . ., V k such that each V i is an independent set and the condition ||V i | − |V j || ≤ 1 holds for every pair i, j , then G is said to be equitably k-colorable.The smallest integer k for which G is equitable k-colorable is known as the equitable chromatic number 5-10 of G and denoted by χ G .Additional graph theory terminology used in this paper can be found in 3, 4 .

Equitable Coloring on Total Graph of Complete Bigraphs
Theorem 2.1.If m ≤ n, the equitable chromatic number of total graph of complete bigraphs K m,n , Proof.Let X, Y be the bipartition of K m,n , where By the definition of total graph, T K m,n has the vertex set {v i : Case 1 if m n, χ T K m,n n 2 .Now we partition the vertex set V T K m,n as follows:

2.2
Clearly V 1 , V 2 , . . ., V n 2 are independent sets and Case 2 if m < n, χ T K m,n n 1 .Now we partition the vertex set V T K m,n as follows:

Equitable Coloring on Central Graph of Cycles and Paths
Theorem 3.1.If n ≥ 5, the equitable chromatic number of central graph of cycles C n , . ., e n } be the vertices and edges of C n taken in the cyclic order.By the definition of central graph, C C n has the vertex set V C n ∪ {u i : 1 ≤ i ≤ n}, where u i is the vertex of subdivision of the edge e i and joining all the nonadjacent vertices of Case 2 n is even .Now we partition the vertex set V C C n as follows: u n−5 , u n−4 }.Clearly V 1 , V 2 , . ..V n/2 are independent sets of C P n .Also |V 1 | |V 3 | |V 4 | • • • |V n/2 | 4 and |V 2 | 3. The inequality V i | − |V j ≤1 holds for any i / j, χ C P n ≤ n/2.For each i, v i is nonadjacent with v i−1 and v i 1 and hence χ C P n ≥ n/2.That is, χ C P n ≥ χ C P n ≥ n/2, χ C P n ≥ n/2.Therefore, χ C P n n/2.Remark 3.4.If n 1, 2, 3, 4, then χ C P n 1, 2, 3, 3, respectively.