The edge fault-tolerance about the strong Menger edge-connectivity of order r among hamming graph

Let r >= 0 be a fixed integer. If F is an edge set of a connected graph G satisfying the minimum degree of G-F being at least r, then F is a conditional faulty edge set of order r. The graph G is called F-strongly Menger-edge-connected if each pair of vertices u and v are connected by min{deg(G-F) (u), deg(G-F) (v)} edge-disjoint paths in G-F, where deg(G-F) (u) and degG-F (v) are the degrees of u and v in G-F, respectively. A graph G is m-strongly Menger-edge-connected of order r if G is F-strongly Menger-edge-connected of order r for every F subset of E(G) with vertical bar F vertical bar = m and F is a conditional edge fault set of order r, and the maximum value of m is written as smr(lambda)(r)(G). Hamming graph K-L(n) has been widely concerned by researchers due to its excellent properties such as good connectivity, scalability, symmetry and iterativity. This paper considers various sufficient conditions of K-L(n) to be F-strongly Menger-edge-connected of order r and determine the exact value of sm(lambda)((L-1)r). (K-L(n)) by studying the edge-disjoint paths in K-L(n) with edge faults, sm(lambda)((L-1)r). (K-L(n)) = (L-1)L-r (n-r)-(L-1)n, where 0 <= r <= n-2 and n >= 3. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.