The 2-path-bipanconnectivity of hypercubes

In this paper, we introduce the concept of the k-path-(bi)panconnectivity of (bipartite) graphs. It is a generalization of the (bi)panconnectivity and of the paired many-to-many k-disjoint path cover. The 2-path-bipanconnectivity with only one exception of the n-cube Qn (n ⩾ 4) is proved. Precisely, the following result is obtained: In an n-cube with n ⩾ 4 given any four vertices u1, v1, u2, v2 such that two of them are in one partite set and the another two are in the another partite set. Let s = t = 5 if C = u1u2v1v2 is a cycle of length 4, and s = d(u1, v1) + 1 and t = d(u2, v2) + 1 otherwise, where d(u, v) denotes the distance between two vertices u and v. And let i and j be any two integers such that both i − s ⩾ 0 and j − t ⩾ 0 are even with i + j ⩽ 2n. Then there exist two vertex-disjoint (u1, v1)-path P and (u2, v2)-path R with ∣V(P)∣ = i and ∣V(R)∣ = j. As consequences, many properties of hypercubes, such as bipanconnectivity, bipanpositionable bipanconnectivity , bipancycle-connectivity , two internally disjoint paths with two given lengths, and the 2-disjoint path cover with a path of a given length , follow from our result.