Bipanconnectivity of balanced hypercubes

The balanced hypercube, proposed by Wu and Huang, is a variant of the hypercube network. In this paper, paths of various lengths are embedded into balanced hypercubes. A bipartite graph GG is bipanconnected if, for two arbitrary nodes xx and yy of GG with distance d(x,y)d(x,y), there exists a path of length ll between xx and yy for every integer ll with d(x,y)≤l≤|V(G)|−1d(x,y)≤l≤|V(G)|−1 and l−d(x,y)≡0l−d(x,y)≡0 (mod 2). We prove that the nn-dimensional balanced hypercube BHnBHn is bipanconnected for all n≥1n≥1. This result is stronger than that obtained by Xu et al. which shows that the balanced hypercube is edge-bipancyclic and Hamiltonian laceable.