The(d,k)-dominating number is a new measure to characterize reliability of resource- sharing in fault tolerant networks.This paper obtains that the(n,2n)-dominating number of the n-dimensional undirected toroidal mesh C(3,3,…,3)is equal to 3(n≥3).
n阶连通图G的路由选择R是由连接G的每个有向顶点对的n(n-1)条路组成.R经过G的每个顶点(每条边)的路的最大条数称为G关于R的点转发指数ξ(G,R)(边转发指数π(G,R)).对G的所有路由选择R,ξ(G,R)(π(G,R))的最小值称为G的点转发指数ξ(G)(边转发指数π(G)).对于k正则k连通图G,Fernandez de la Vega和Manoussakis[Discrete Applied Mathematics,1989,23(2) :103-123]证明ξ(G)≤(n-1)·┌(n-k-1)/k┐和π(G)≤n┌(n-k-1)/k┐,并且猜想ξ(G)≤┌(n-k)(n-k-1)/k┐.我们分别改进了ξ(G)≤(n-1) ┌(n-k-1)/k┐-(n-k-1)和π(G)≤n┌(n-k-1)/k┐-(n-k),并且证明了猜想对k=3的情形.
非空图G的约束数b(G)是指使得图G的控制数γ(G)增大而删除的最少的边数.[Fischermann M,Rautenbach D,Volkmann L.Remarks on the bondage number of planar graphs.Discrete Math,2003,260:57-67]已经证明,对于一个围长为g(G)的平面图G,如果g(G)≥4则b(G)≤6,如果g(G)≥5则b(G)≤5,如果g(G)≥6则b(G)≤4,如果g(G)≥8则b(G)≤3.我们把这个结果推广到连通的超环面图中.
