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The balanced hypercube, BH n , is a variant of hypercube Q n . Zhou et al. [Inform. Sci. 300 (2015) 20-27] proposed an interesting problem that whether there is a fault-free Hamiltonian cycle in BH n with each vertex incident to at least two fault-free edges. In this paper, we consider this problem and show that each fault-free edge lies on a fault-free Hamiltonian cycle in BH n after no more than 4n − 5 faulty edges occur if each vertex is incident with at least two fault-free edges for all n ≥ 2. Our result is optimal with respect to the maximum number of tolerated edge faults.

A k-container C(u, v) of a graph G is a set of k internally disjoint paths between u and v. A k-container C(u, v) of G is a k * -container if it is a spanning subgraph of G. A graph G is k *connected if there exists a k * -container between any two different vertices of G. A k-regular graph G is super spanning connected if G is i * -container for all 1 ≤ i ≤ k. In this paper, we prove that the arrangement graph A n,k is super spanning connected if n ≥ 4 and n − k ≥ 2.

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