βee0-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 β0ee(G) and an externally equitable and independent set with cardinality β0ee(G) is called a β0ee(G)-set of G. A vertex u is said to be β0ee(G)-good if u belongs to a β0ee(G)-set of G. If every vertex of G is β0ee(G)-good, then G is said to be β0ee-excellent graph. In this paper, this concept is introduced and studied.