The cartesian product digraph is one of the most important models for large-scale interconnected networks. As a common generalization of the arc connectivity and the restricted arc connectivity ,the k-restricted arc connectivity of digraphs can be used to measure the reliability of networks. Let D be a strong digraph. An arc subset S is a k-restricted arc that cut of D if D-S has a strong component D1 with order at least k such as D-V(D1) contains a connected subdigraph with order at least k. If the k-restricted arc cut exists,then D is called λk-connected. The k-restricted arc connectivityλk(D1)of aλk-connected digraph D is the minimum cardinality over all k-restricted arc cuts. In this paper,we show an upper bound on the k-restricted arc connectivity of cartesian product digraphs and a lower bound on the 3-restricted arc connectivity of cartesian product digraphs,which extends the results of 2-restricted arc connectivity. Furthermore,we give an example to show that these bounds are sharp.%笛卡尔积图是大型互联网络最重要的数学模型之一.有向图的k-限制弧连通度是弧连通度和限制弧连通度的推广,可用于度量网络的可靠性.强连通有向图D的弧子集S被称为D的一个k-限制弧割,若D-S有一个顶点数至少为k的强连通分支D1,使得D-V(D1)包含一个顶点数至少为k的连通子图.若这样的一个弧割存在,则称D是λk-连通的.D中最小k-限制弧割所含的弧数称为D的k-限制弧连通度,记做λk(D).在有向笛卡尔积图中,推广2-限制弧连通度的结论到k-限制弧连通度,得到有向笛卡尔积图的k-限制弧连通度的上界和3-限制弧连通度的下界,并用例子说明所得界是紧的.
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