3-extra Connectivity of K-ary N-cube Networks

Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G − S is disconnected and each component of G− S has at least h + 1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h = 1, 2 of k-ary n-cube are gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k ≥ 4 and Zhu et al. in [Theory of Computing Systems, arxiv.org/pdf/1105.0991v1 [cs.DM] 5 May 2011] for k = 3. In this paper, we show that the h-extra connectivity of the 3-ary n-cube networks for h = 3 is equal to 8n− 12, where n ≥ 3.