A note on the conditional fault-tolerant strong Menger edge connectivity of regular graphs

Abstract

A graph G is said to be strongly Menger edge-connected (SM-λ for short) if for every two of its vertices u and v, there are min{dG(u),dG(v)}) edge-disjoint paths between them, where dG(u) is the degree of u in G. The maximum conditional edge-fault-tolerant number of order r with respect to the SM-λ property of G, denoted by smλr(G), is the maximum integer m such that G−F remains SM-λ for every edge set F with |F|≤m and δ(G−F)≥r. So far, most of the exact values of smλr(G) are for the case r≤2 for special class of graphs. In this paper, we give a sufficient condition to give a bound of smλr(G) for general r on regular graphs. As an application, we determine the exact value of smλ2r−1(AQn) for 1≤r≤n−2 where AQn is the n-dimensional augmented cube.