3-extra connectivity of 3-ary n-cube networks

Abstract: 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.