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 .