A subset F ⊂ V (G) is called an R^{k}-vertex-cut of a graph G if G -F is disconnected and each vertex of G -F has at least k neighbors in G -F. The R^{k}-vertex-connectivity of G, denoted by κ^{k}(G), is the cardinality of a minimum R^{k}-vertex-cut of G. Let B_{n} be the bubble sort graph of dimension n. It is known that κ_{k}(B_{n})=2^{k}(n -k -1) for n ≥ 2k and k=1, 2. In this paper, we prove it for k=3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.
In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.