Edge-fault-tolerant strong Menger edge connectivity on regular graphs

Abstract

The connectivity of a graph is an important issue in graph theory and is also one of the most important factors in evaluating the reliability and fault tolerance of a network. A graph G is called m-edge-fault-tolerant strongly Menger (m-EFTSM for short) edge connected if there are min⁡{degG−F⁡(x),degG−F⁡(y)} edge-disjoint paths between any two different vertices x and y in G−F for any F⊆E(G) with |F|≤m.In this paper, we give a necessary and sufficient condition of EFTSM edge connectivity on regular graphs. And we obtain several optimal results about EFTSM edge connectivity on (1, 2)-matching composition networks, each of which is constructed by connecting two graphs via one or two perfect matchings. As applications, we show that the class of n-dimensional hypercube-like networks (included hypercube, crossed cube et al.) are (n−2)-EFTSM edge connected; show that the n-dimensional folded hypercube is (n−1)-EFTSM edge connected, and show that the n-dimensional augmented cube is (2n−3)-EFTSM edge connected. The bounds (n−2),(n−1) and (2n−3) are sharp.