Fault-tolerant analysis of augmented cubes

The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\ge 4)$. This paper determines that the 2-extra connectivity of $AQ_n$ is $6n-17$ for $n\geq 9$ and the 2-extra edge-connectivity is $6n-9$ for $n\geq 4$. That is, for $n\geq 9$ (respectively, $n\geq 4$), at least $6n-17$ vertices (respectively, $6n-9$ edges) of $AQ_n$ have to be removed to get a disconnected graph that contains no isolated vertices and isolated edges. When the augmented cube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system

Panpositionable hamiltonicity of the alternating group graphs, Networks 50

The alternating group graph AG n is an interconnection network topology based on the Cayley graph of the alternating group. There are some interesting results concerning the hamiltonicity and the fault tolerant hamiltonicity of the alternating group graphs. In this article, we propose a new concept called panpositionable hamiltonicity. A hamiltonian graph G is panpositionable if for any two different vertices x and y of G and for any integer l satisfying d (x , y ) ≤ l ≤ |V (G)| − d (x , y ), there exists a hamiltonian cycle C of G such that the relative distance between x , y on C is l . We show that AG n is panpositionable hamiltonian if n ≥ 3