Dominating cocoloring of graphs

Abstract

A -cocolouring of a graph G is a partition of the vertex set into subsets such that each set induces either a clique or an independent set in G. The cochromatic number Z(G)of a graph G is the least such that has a -cocolouring of G. A set S⊆V is a dominating set of G if for each u∈V, there exists a vertex V∈S such that is adjacent to V. The minimum cardinality of a dominating set in is called the domination number and is denoted by γ(G). Combining these two concepts we have introduces two new types of cocoloring viz, dominating cocoloring and γ-cocoloring. A dominating cocoloring of is a cocoloring of such that atleast one of the sets in the partition is a dominating set. Hence dominating cocoloring is a conditional cocoloring. The dominating co-chromatic number is the smallest cardinality of a dominating cocoloring of G.(ie)Zd=min{k\g has a dominating cocoloring with k-colors.