A set of spanning trees of a graphs G are called completely independent spanning trees (CISTs for short) if for every pair of vertices x, y∈V(G), the paths joining x and y in any two trees have neither vertex nor edge in common, except x and y. Constructing CISTs has applications on interconnection networks such as fault-tolerant routing and secure message transmission. In this paper, we investigate the problem of constructing two CISTs in the balanced hypercube BHn, which is a hypercube-variant network and is superior to hypercube due to having a smaller diameter. As a result, the diameter of CISTs we constructed equals to 9 for BH2 and 6n-2 for BHn when n≥3.

Let T1,T2,...,Tk be spanning trees in a graph G. If, for any two vertices u,v of G, the paths joining u and v on the k trees are mutually vertex-disjoint, then T1,T2,...,Tk are called completely independent spanning trees (CISTs for short) of G. The construction of CISTs can be applied in fault-tolerant broadcasting and secure message distribution on interconnection networks. Hasunuma (2001) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Unfortunately, this conjecture was disproved by Péterfalvi recently. In this note, we give a necessary condition for k-connected k-regular graphs with ⌊k/2⌋ CISTs. Based on this condition, we provide more counterexamples for Hasunuma's conjecture. By contrast, we show that there are two CISTs in 4-regular chordal rings CR(N,d) with N=k(d-1)+j under the condition that k ≥ 4 is even and 0 ≤ j ≤ 4. In particular, the diameter of each constructed CIST is derived.

A queue layout of a graph G consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The queuenumber qn(G) is the minimum number of queues required in a queue layout of G. The Cartesian product of two graphs G1 = (V1,E1) and G2 = (V2,E2), denoted by G1 × G2, is the graph with {:v1 ∈ V1 and v2 ∈ V2} as its vertex set and an edge (,) belongs to G1×G2 if and only if either (u1,v1) ∈ E1 and u2 = v2 or (u2,v2) ∈ E2 and u1 = v1. Let Tk1,k2,...,kn denote the n-dimensional toroidal grid defined by the Cartesian product of n cycles with varied lengths, i.e., Tk1,k2,...,kn = Ck1 × Ck2 × … × Ckn, where Cki is a cycle of length ki ≥ 3. If k1 = k2 = … = kn = k, the graph is also called the k-ary n-cube and is denoted by Qnk. In this paper, we deal with queue layouts of toroidal grids and show the following bound: qn(Tk1,k2,...,kn) ≤ 2n-2 if n ≥ 2 and ki ≥ 3 for all i = 1,2,...,n. In particular, for n = 2 and k1,k2 ≥ 3, we acquire qn(Tk1,k2) = 2. Recently, Pai et al. (Inform. Process. Lett. 110 (2009) pp.50-56) showed that qn(Qnk) ≤ 2n-1 if n ≥1 and k ≥9. Thus, our result improves the bound of qn(Qnk) when n ≥2 and k ≥9.

Given a graph G, a set of spanning trees of G are completely independent spanning trees (CISTs for short) if for any vertices x and y, the paths connecting them on these trees have neither vertex nor edge in common, except x and y. Hasunuma (2001, 2002) first introduced the concept of CISTs and conjectured that there are k CISTs in any 2k-connected graph. Later on, this conjecture was unfortunately disproved by Péterfalvi (2012). In this note, we show that Hasunuma's conjecture holds for graphs restricted in the class of 4-regular chordal rings CR(n,d), where both n and d are even integers.