Strongly Menger connectedness of data center network and (, )-star graph.

A graph G=(V,E) is F-strongly Menger-vertex-connected (resp. Menger-edge-connected), if for a subgraph G−F of G with a conditional faulty vertex (resp. edge) set F⊆V (resp. F⊆E), each pair of vertices u and v are connected by min{degG−F(u),degG−F(v)} internally disjoint (resp. edge-disjoint) fault-free paths in G−F. Where degG−F(u) and degG−F(v) are the degrees of u and v in G−F, respectively. He et al. (2018) [7] introduced some sufficient conditions of F-strongly Menger (edge) connected for k-regular triangle-free graphs. In this paper, we consider the strongly Menger connectedness of two kinds of graphs with triangles: data center network Dk,n and (n,k)-star graph Sn,k. We prove Dk,n is (k−1)-strongly Menger-vertex-connected, (n+k−3)-strongly Menger-edge-connected of order 1, (2n+2k−8)-strongly Menger-edge-connected of order 2, and (3n+3k−15)-strongly Menger-edge-connected of order 3. Furthermore, we obtain Sn,k is (k−2)-strongly Menger-vertex-connected, (n−3)-strongly Menger-edge-connected of order 1, (2n−8)-strongly Menger-edge-connected of order 2, and (3n−15)-strongly Menger-edge-connected of order 3.