Study of Graph Partitions Approach satisfies Vizing's Conjecture

Abstract

For a graph G(V,E) , a subset of vertices D is a dominating set if for each vertex 12x∈"> V either  12x∈D">  , or x is adjacent to at least one vertex of D . The domination number , 12خ³G, "> is the order of smallest dominating set of G . In [7],  Vizing conjectured that 12خ³Gأ—H ≥ خ³Gأ—خ³H">  for any two graphs G and H , where G×H denotes their Cartesian product . This conjecture is still open .

In this paper , we investigate following relations, if a graph H has a D-partition then it also has a K-partition, and if  H  has a K-partition , then Vizing's conjecture is satisfied for any graph G , after that, every cycle 12Cn , n≥3">  , has a K-partition. Moreover, if H has    a K-partition , then  H satisfies the following relations  12خ³H≤2"> , 12P2H=خ³H">  and H is a perfect-dominated graph .