Regular connected bipancyclic spanning subgraphs of hypercubes

An n -dimensional hypercube Q n is a Hamiltonian graph; in other words Q n ( n ≥ 2 ) contains a spanning subgraph which is 2-regular and 2-connected. In this paper, we explore yet another strong property of hypercubes. We prove that for any integer k with 3 ≤ k ≤ n , Q n ( n ≥ 3 ) contains a spanning subgraph which is k -regular, k -connected and bipancyclic. We also obtain the result that every mesh P m × P n ( m , n ≥ 2 ) is bipancyclic, which is used to prove the property above.