Complementary Degree Equitable Sets in Graphs

Abstract

Let $G=(V,E)$ be a simple graph. The concept of equitability was first introduced by W.Meyer.\cite{Mew} . A proper colouring is called equitable if the cardinalities of any two colour classes differ by one. Inspired by this concept, E.Sampathkumar formulated degree equitability. Two vertices in a simple graph are said to be degree equitable if the difference of the degrees of the two vertices differ by one. This concept of degree equitability was used to develop degree equitable sets, degree equitable domination and its variations and equitable colouring in \cite{Ani1,Ani2,vsn,Lak,muthu}. A subset $S$ is said to be complementary equitable if for any $u,v \in V-S, |deg(u)-deg(v)| \leq 1$. In this paper, a study of complementary degree equitable sets is initiated.