The super connectivity of augmented cubes

Abstract

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 = 3). This paper determines that the super connectivity of AQ n is 4n − 8 for n 6 and the super edge-connectivity is 4n − 4 for n 5. That is, for n 6 (respectively, n 5), at least 4n − 8 vertices (respectively, 4n − 4 edges) of AQ n are removed to get a disconnected graph that contains no isolated vertices. 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.

A note on “The super connectivity of augmented cubes”

“…The proof strongly depends on whether R − F R is connected or not. However, in [1], we only showed this conclusion for the former, and neglected the later. Now, we replenish our proof.…”

“…It is clear that ξ(AQ n ) = 4n − 4. By Lemma 2 in [1], we only need to prove λ (AQ n ) 4n − 4 for n 2. The proof proceeds by induction on n 2.…”

Conditional edge-fault Hamiltonicity of augmented cubes☆

“…Several topological properties of AQ n have been investigated [7,23,27,32,33]. Ma et al [33] studied the super connectivity of AQ n .…”

Edge-fault-tolerant vertex-pancyclicity of augmented cubes

“…Moreover, AQ n possesses some embedding properties that Q n does not. Previous works relating to the augmented hypercube can be found in [2,5,[11][12][13]16,17,21,24].…”