Many-to-Many Disjoint Paths in Augmented Cubes With Exponential Fault Edges

Abstract

It is common knowledge that edge disjoint paths have close relationship with the edge connectivity. Motivated by the well-known Menger theorem, we find that the maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of G (λ g (G)). It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The n-dimensional augmented cube AQ n is a variant of hypercube Q n . In this paper, we observe that the g-extra edge-connectivity of the augmented cube for some exponentially large enough g exists a concentration behavior, for about 72.22 percent values of g ≤ 2 n−1 , and that the g-extra edge-connectivity of AQ n (n ≥ 3) concentrates on n 2 − 1 special values. Specifically, we prove that the exact value of g-extra edge-connectivity of augmented cube is a constant 2( n 2 − r)2+r , where n ≥ 3, r = 1, 2, . . . , n 2 − 1 and l r = 2 2r+1 −2 3 if n is odd and l r = 2 2r+2 −4 3 if n is even. The above upper and lower bounds of g are sharp. Moreover, we also obtain the exponential edge disjoint paths in AQ n with edge faults.INDEX TERMS Fault tolerance, many-to-many edge disjoint paths, interconnected networks, exponential fault edges.