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 Q"n (n>=4) is proved. Precisely, the following result is obtained: In an n-cube with n>=4 given any four vertices u"1, v"1, u"2, v"2 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=u"1u"2v"1v"2 is a cycle of length 4, and s=d(u"1,v"1)+1 and t=d(u"2,v"2)+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=<2^n. Then there exist two vertex-disjoint (u"1,v"1)-path P and (u"2,v"2)-path R with |V(P)|=i and |V(R)|=j. As consequences, many properties of hypercubes, such as bipanconnectivity, bipanpositionable bipanconnectivity [18], bipancycle-connectivity [12], two internally disjoint paths with two given lengths, and the 2-disjoint path cover with a path of a given length [21], follow from our result.