b-CHROMATIC NUMBER OF CORONA PRODUCT OF CROWN GRAPH AND COMPLETE BIPARTITE GRAPH WITH PATH GRAPH

A b-coloring of a graph is a proper coloring where each color admits at least one node (called dominating node) adjacent to every other used color. The maximum number of colors needed to b-color a graph G is called the b-chromatic number and is denoted by φ(G). In this paper, we find the b-chromatic number and some of the structural properties of corona product of crown graph and complete bipartite graph with path graph.


INTRODUCTION
A b-coloring by k-colors is a proper coloring of the vertices of graph G such that in each color classes there exists a vertex that has neighbors in all the other k-1 color classes. The b-chromatic number φ(G) is the largest number k for which G admits a bcoloring with k-colors (Irving and Manlove, 1999). The corona G1 • G2 of two graphs G1 and G2 is defined as a graph obtained by taking one copy of G1 (which has p1 vertices) and p1 copies of G2 and attach one copy of G2 at every vertex of G1 (Harary, 1972).
In this paper we find for which the largest number k for which corona product of crown graph and complete bipartite graph with path graph admits a b-coloring with k-colors. And also we find some of its structural properties (Venkatachalam and Vernold Vivin, 2010;Vernold Vivin and Venkatachalam, 2012;Vijayalakshmi and Thilagavathi, 2012)

Crown Graph
A crown graph on 2n vertices is an undirected graph with two sets of vertices ui and vi and with an edge from ui to vj whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed (Wikipedia).

Complete Bipartite Graph
A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V1 and V2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph (V1, V2, E) such that for every two vertices v1 ∈ V1 and v2 ∈ V2, v1v2 is an edge in E. A complete bipartite graph with partitions of size |V1|=m and |V2|=n, is denoted Km,n (Balakrishnan, 2004;Balakrishnan and Ranganathan, 2012).

and
A Fan graph Fm,n is defined as the graph join Km , where Km the empty graph on nodes is and Pn is the path on n nodes (Wikipedia).

Path Graph
The path graph Pn is a tree with two nodes of vertex degree 1, and the other n-2 nodes of vertex degree (Harary, 1972).

Corona Product
Corona product or simply corona of any graph G1 and graph G2, defined as the graph which is the disjoint union of one copy of G1 and |V1| copies of G2 (|V1| is the number of vertices of G1) in which each vertex of the copy of G1 is connected to all vertices of a separate copy of G2 (Harary, 1972).

b-coloring
A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by φ (G), is the maximum number t such that G admits a b-coloring with t colors (Irving and Manlove, 1999). n =1 + (2 3
Let be ant path graph of length n-1 with n-vertices. V( By the definition of corona graph each vertex in S 0 is adjacent to every vertex copy of , i.e. vertices of ( Consider the color class C = { 1, 2, , 3, … , , +1, +2, … , 2 } to color the vertices of ( 0 • ). Assign the colors

Illustration Corona Product of Complete
the above coloring to be b-chromatic proper coloring of V( ) by corresponding non-adjacent vertices of its 's or , ′ respectively,and the remaining vertices are colored properly by the colors in the color class. Thus each color has the neighbor in the every other color class.

CONCLUSION
In this paper we operated the graph operation corona product on crown graph and complete bipartite graph with path graph, we get corona product of crown graph with path graph and corona product of complete bipartite graph with path graph and also we find its b-chromatic number and some of its structural properties.