β ee0-excellence in graphs

The concept of equitability was first introduced by W. Meyer in his paper titled Equitable Coloring[4]. In this paper, cardinality equitability between color classes was considered. Prof. E. Sampathkumar introduced the concept of degree equitability for vertices in a graph. Several types of equitability have been considered[1][2]. External equitability has been introduced in [3]. A subset T of the vertex set V of a graph G , one can define externally equitable, that is, if for any x,y in the complement of T , ‖ N ( x )∩ T |− N ( y )∩ T ‖ ≤ 1. A subset S of the vertex set of a graph is externally equitably independent if S is independent and S is externally equitable. The maximum cardinality of an externally equitable and independent set is denoted by β 0 e e ( G ) and an externally equitable and independent set with cardinality β 0 e e ( G ) is called a β 0 e e ( G ) -set of G . A vertex u is said to be β 0 e e ( G ) -good if u belongs to a β 0 e e ( G ) -set of G . If every vertex of G is β 0 e e ( G ) -good, then G is said to be β 0 e e -excellent graph. In this paper, this concept is introduced and studied.