• COLORFUL TOTAL DOMINATION WITH RESPECT TO COLOR CLASS DOMINATION PARTITION

Abstract

Let G be a finite, simple and undirected graph. A partition  of  V(G)  into  independent  sets such that  each element   of the  partition  is dominated  by a   vertex  of  G  is called  color class domination  partition of G  [1],[2],[6],[7]. The minimum cardinality of such a partition is called the color class domination partition number of G and is denoted by (G).  If S is a  -partition of  G,  then the set  consisting of  one vertex  from each element of the partition need not  be a dominating set.  Since Π = {{u₁},{u₂},....., {u_n}} where V(G) ={u₁,u₂,....., u_n} is a  cd-partition  of  G  such that the  set  consisting of one element from each partition  is a dominating  set of G. The  minimum  cardinality  of a   cd-partition  in which  the set  consisting one  element from  each  set of the partition  is a dominating  set  is called   the colorful domination number of G with respect to  cd-partition of  G and is denoted by (G). If G has no isolates, then the trivial partition is a cd-partition which gives rise to a total dominating set of G. The minimum cardinality of a cd-partition which gives rise to a total dominating set of G is called the colorful total domination number of G and is denoted by . In this paper, a study of this new parameter is initiated. (G  for  well  known graphs are found,  bounds are obtained  and  for bipartite  graph  in certain conditions,  the value of    is  obtained and bound  for   is also  derived.