Domination Parameters And Hamilton Decomposition Of Cycle Product On Generalized Base-b Hypercube Networks

Interconnection network is an important part of super computers.Its topological structure refers to the very large scale in the computer system components(proces-sor)connection mode.In fact,its topological structure is a graph.The structure and properties of interconnection network are important research projects of the super computer.When you design and select a topological structure for an interconnection network(some indicators such as degree of vertex,Hamiltonian,connectivity and diameter play an important role in analyzing the performance of interconnection network.In this paper,we discussed the domination numbers of generalized base-b hyper-cube and Cartesian product graph GCn(b)×H,the total domination number,the independent domination number,the connected domination number,the perfect domination number,the signed domination number and the inverse signed domina-tion number of the generalized base-b hypercube,Hamilton decomposition of the Cartesian product network GCn(b)×Cm1×Cm2×…×Cmq.Which obtains the following results:1.The main res,ults of generalized base-b hypercube networks:In 1993,Gen-eralized base-b hypercube has been put forward by S.Lakshmivarahan,J.S.J-wo,S.K.Dhall:which is famous interconnection network.In this paper,(1)we s-tudy the exact values of the domination numbers of generalized base-b hypercube for 1<n?5,b = 3;1<n?3,b= 4 and the bounds of the domination numbers for n?5,b = 3;n?4,b=4.(2)we study the exact values of the total domination numbers of generalized base-b hypercube for 1<n?5.b=3;1<n<3,b=4 and the bounds of the total domination numbers for n?6,6=3;n?4,b=4.(3)we study the exact values of the independent domination numbers of generalized base-b hypercube for 1<n?5,b=3;1<n?3,b = 4 and the bounds of the independent domination numbers for n?4,b=3;n?4,b=4.(4)we study the exact values of the connected domination numbers of generalized base-b hyper-cube for 1<n?4,b = 3 and the bounds of the connected domination numbers for 5?n?7,b = 3.(5)we study the exact values of the perfect domination numbers of generalized base-b hypercube for 1<n?4,b = 3 and the bounds of the perfect domination numbers for n?5,b = 3.(6)we study the exact values of the signed domination numbers and the inverse signed domination number of generalized base-b hypercube for 2?n?4,b= 2,b = 4 and the bound-s of the signed domination numbers and the inverse signed domination number for n?2,b = 3;n?2,b = 4 and the bounds of the inverse signed domination number for n?2,b?3.2.The main results of the Cartesian product graph GCn(b)×H:In 1963,Viz-ing proposed the famous Vizing conjecture:for any two graphs G and H,there are:?(G × H)??(G)?(H).Haizhong Shi thinks that GCn(b)satisfies Vizing con-jecture.In this paper,we proved that 1?n?6,b = 2,3,the graph GCn(b)satisfies conjecture.3.The main results of the Cartesian product network GCn(b)×Cm1×Cm2×…×Cmq.Haizhong Shi designed an interconnection network-the Cartesian prod-uct network GCn(b)×Cm1×Cm2×…Cmq,And Haizhong Shi proposed the following conjecture:the cartesian product network GCn(b)×Cm1×Cm2×…×Cmq can be Hamilton decomposed,among them,the GCn(b)is generalized base-b hypercube,Cmj is mj long cycle,j=1,2,…,q.In this paper,it is proved that when n=1,1?b?6,q = 1,m1 = 3,4;n= 2,b=2,3,q = 1,m1= 3,4 can be Hamilton decomposed.