Edge fault-tolerance of strongly Menger edge connected graphs

A connected graph G is strongly Menger edge connected (SM-λ for short) if any two of its vertices x,y are connected by min⁡{d(x),d(y)} edge-disjoint paths, where d(x) is the degree of x. The maximum edge-fault-tolerant with respect to the SM-λ property of G, denoted by smλ(G), is the maximum integer m such that G−F is still SM-λ for any edge-set F with |F|≤m. In this paper, we give a sharp lower and upper bound for smλ(G), and give a characterization of the minimum upper bound. Furthermore, for k-regular graphs, we give some examples to show that smλ(G) can reach every value between the lower bound and the upper bound when k is odd; show that smλ(G) can reach every even value between the lower bound and the upper bound, and the only possible odd value is k−1 if k is even. Moreover, we completely determine the exact value of smλ(G) when G is a vertex-transitive graph or when it is an edge-transitive graph.