In this paper, we introduce a concise representation, called right-distance sequences (or RD-sequences for short), to describe all t-ary trees with n internal nodes. A result reveals that there exists a close relationship between the representation and the well-formed sequences suggested by Zaks [Lexicographic generation of ordered trees, Theoretical Computer Science 10 (1980) 63-82]. Using a coding tree and its concomitant tables, a systematical way can help us to investigate the structural representation of t-ary trees. Consequently, we develop efficient algorithms for determining the rank of a given t-ary tree in lexicographic order (i.e., a ranking algorithm), and for converting a positive integer to its corresponding RD-sequence (i.e., an unranking algorithm). Both the ranking and unranking algorithms can be run in O(tn) time and without computing all the entries of the coefficient table.

A non-regular tree T with a prescribed branching sequence (s1,s2,...,sn) is a rooted and ordered tree such that its internal nodes are numbered from 1 to n in preorder and every internal node i in T has si children. Recently, Wu et al. (2010) introduced a concise representation called RD-sequences to represent all non-regular trees and proposed a loopless algorithm for generating all non-regular trees in a Gray-code order. In this paper, based on such a Gray-code order, we present efficient ranking and unranking algorithms of non-regular trees with n internal nodes. Moreover, we show that the ranking algorithm and the unranking algorithm can be run in O(n2) time and O(n2+nSn-1) time, respectively, provided a preprocessing takes O(n2Sn-1) time and space in advance, where .

Given a graph G, a set of spanning trees of G are completely independent 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. In this paper, we prove that for graphs of order n, with n ≥ 6, if the minimum degree is at least n-2, then there are at least ⌊n/3⌋ completely independent spanning trees.

For k ≥ 3, a convex geometric graph is called k-locally outerplanar if no path of length k intersects itself. In [D. Boutin, Convex Geometric Graphs with No Short Self-intersecting Path, Congressus Numerantium 160 (2003) 205-214], Boutin stated the results of the degeneracy for 3-locally outerplanar graphs. Later, in [D. Boutin, Structure and Properties of Locally Outerplanar Graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 60 (2007) 169-180], a structural property on k-locally outerplanar graphs was proposed. These results are based on the existence of “minimal corner pairs”. In this paper, we show that a “minimal corner pair” may not exist and give a counterexample to disprove the structural property. Furthermore, we generalize the result on the degeneracy with respect to k-locally outerplanar graphs.

Generalized recursive circulant graphs (GRCGs for short) are a generalization of recursive circulant graphs and provide a new type of topology for interconnection networks. A graph of n vertices is said to be s-pancyclic for some $3leqslant sleqslant n$ if it contains cycles of every length t for $sleqslant tleqslant n$. The pancyclicity of recursive circulant graphs was investigated by Araki and Shibata (Inf. Process. Lett. vol.81, no.4, pp.187-190, 2002). In this paper, we are concerned with the s-pancyclicity of GRCGs.

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.

An n-dimensional folded hypercube, denoted by FQn, is an enhanced n-dimensional hypercube with one extra link between nodes that have the furthest Hamming distance. Let FFv (respectively, FFe) denote the set of faulty nodes (respectively, faulty links) in FQn. Under the assumption that every fault-free node in FQn is incident to at least two fault-free links, Hsieh et al. (Inform. Process. Lett. 110 (2009) pp.41-53) showed that if |FFv|+|FFe| ≤ 2n-4 for n ≥ 3, then FQn-FFv-FFe contains a fault-free cycle of length at least 2n-2|FFv|. In this paper, we show that, under the same conditional fault model, FQn with n ≥ 5 can tolerate more faulty elements and provides the same lower bound of the length of a longest fault-free cycle, i.e., FQn-FFv-FFe contains a fault-free cycle of length at least 2n-2|FFv| if |FFv|+|FFe| ≤ 2n-3 for n ≥ 5.

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.

Left-weight sequences (LW-sequences for short) are in common currency for encoding binary trees. In [16], Wu et al. proposed an algorithm associated with tree rotations for listing all binary trees in diverse representations including LW-sequences. In particular, such a list of LW-sequences is generated in Gray-code order. In this paper, based on this ordering, we present efficient ranking and unranking algorithms. For binary trees with n internal nodes, the time complexity and the space requirement in each of our ranking and unranking algorithms are O(n2) and O(n), respectively.