Several Kinds Of Dominations And Cycle Properties Of Balanced Hypercube

As a variant of hypercube,balanced hypercube has features not found in other hypercube variants,one of its features is that each processor has a backup processor of the same neighboring nodes.Thus,when a processor fails during a task run,it can be reactivated in its backup processor and continue running.In this paper,several kinds of dominations and cycle properties of balanced hypercube are discussed.The main results are as follows:1.The specific values of domination numbers and total domination numbers of n-dimensional balanced hypercube are determined when n=1,2,3.Professor Haizhong Shi thinks that n-dimensional balanced hypercube satisfies Vizing con-jecture,and this paper studies the product graphs of BH1 with K2,C3 and BH1,respectively,and the product graphs of BH2 with K2,C3 and BH1,respectively.2.The specific values of signed domination numbers,majority domination num-bers and reverse signed domination numbers of n-dimensional balanced hypercube are determined when n=1,2.The specific values of signed domination number and majority domination number and the bound of reverse signed domination number of the n-dimensional balanced hypercube at n=3 are determined.The specific value of majority domination number of n-dimensional balanced hypercube is determined when n=4.3.The specific values of edge domination numbers and signed edge domina-tion numbers of n-dimensional balanced hypercube are determined when n=1,2.The bounds of edge domination number and signed edge domination number of n-dimensional balanced hypercube at n=3 are determined.4.Professor Haizhong Shi proposed the conjecture:The n-dimensional balanced hypercube BHn(n?2)is 2 factorable decomposition with a length of at least k(6?k?4n)for each cycle.In this paper,it is proved that n-dimensional balanced hypercube has a 2 factorable decomposition with each cycle length at least 8 for n=2,3.It is proved that the n-dimensional balanced hypercube has a 2 factorable decomposition with each cycle length of at least 16 for n=4.