Toru Hasunuma

Independent spanning trees with small depths in iterated line digraphs

We show that the independent spanning tree conjecture on digraphs is true if we restrict ourselves to line digraphs. Also, we construct independent spanning trees with small depths in iterated line digraphs. From the results, we can obtain independent spanning trees with small depths in de Bruijn and Kautz digraphs that improve the previously known upper bounds on the depths.

Completely independent spanning trees in the underlying graph of a line digraph

Abstract In this note, we define completely independent spanning trees. We say that T1,T2,…,Tk are completely independent spanning trees in a graph H if for any vertex r of H, they are independent spanning trees rooted at r. We present a characterization of completely independent spanning trees. Also, we show that for any k-vertex-connected line digraph L(G), there are k completely independent spanning trees in the underlying graph of L(G). At last, we apply our results to de Bruijn graphs, Kautz graphs, and wrapped butterflies.

Completely independent spanning trees in torus networks

Let T1, T2, …, Tk be spanning trees in a graph G. If for any two vertices u, v in G, the paths from u to v in T1, T2, …, Tk are pairwise internally disjoint, then T1, T2, …, Tk are completely independent spanning trees in G. Completely independent spanning trees can be applied to fault-tolerant communication problems in interconnection networks. In this article, we show that there are two completely independent spanning trees in any torus network. Besides, we generalize the result for the Cartesian product. In particular, we show that there are two completely independent spanning trees in the Cartesian product of any 2-connected graphs.

Completely Independent Spanning Trees in Maximal Planar Graphs

Let G be a graph. Let T1, T2, . . . , Tk be spanning trees in G. If for any two vertices u, v in G, the paths from u to v in T1, T2, . . . , Tk are pairwise openly disjoint, then we say that T1, T2, . . . , Tk are completely independent spanning trees in G. In this paper, we show that there are two completely independent spanning trees in any 4-connected maximal planar graph. Our proof induces a linear-time algorithm for finding such trees. Besides, we show that given a graph G, the problem of deciding whether there exist two completely independent spanning trees in G is NP-complete.

A linear time algorithm for L(2,1)-labeling of trees

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x) − f(y)| ≥ 2 if x and y are adjacent and |f(x) − f(y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,...,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ4.5 n) for more than a decade, and an O( min {n 1.75,Δ1.5 n})-time algorithm has appeared recently, where Δ is the maximum degree of T and n = |V(T)|, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem for L(2,1)-labeling of trees by establishing a linear time algorithm.

Laying Out Iterated Line Digraphs Using Queues

In this paper, we study a layout problem of a digraph using queues. The queuenumber of a digraph is the minimum number of queues required for a queue layout of the digraph. We present upper and lower bounds on the queuenumber of an iterated line digraph L k (G) of a digraph G. In particular, our upper bound depends only on G and is independent of the number of iterations k. Queue layouts can be applied to three-dimensional drawings. From the result on the queuenumber of L k (G), it is shown that for any fixed digraph G, L k (G) has a three-dimensional drawing with O(n) volume, where n is the number of vertices in L k (G). We also apply these results to particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, butterfly digraphs, and wrapped butterfly digraphs.

Embedding de Bruijn, Kautz and shuffle-exchange networks in books

We show that the de Bruijn digraph B(d, D), D > 1 and the Kautz digraph K(d, D) can be embedded in (d + 1) pages with cumulative pagewidth 14dD − 2(3d3 − 2d2 + 4d − d(d mod 2) − 4) and 14dD − 1(3d2 + 4d + (d mod 2)), respectively. Also we show that the shuffle-exchange graph S(D), D > 2 can be embedded in 3 pages with cumulative pagewidth 5 ·2D − 3. From these results, the pagenumbers of B(2, D), K(2, D) and S(D) are determined to be 3.

Embedding iterated line digraphs in books

In this paper, we present an upper bound on the pagenumber of an iterated line digraph Lk(G) of a digraph G. Our bound depends only on the digraph G and is independent of the number of iterations k. In particular, it is proved that the pagenumber of Lk(G) does not increase with the number of iterations k. This result generalizes previous results on book-embeddings of some particular families of iterated line digraphs such as de Bruijn digraphs, Kautz digraphs, and butterfly networks. Also, we apply our result to wrapped butterfly networks.

Counting small cycles in generalized de Bruijn digraphs

In this paper, we count small cycles in generalized de Bruijn digraphs. Let n = pd h , where d? p, and g l = gcd(d l - 1, n). We show that if p d 3 and k ≤ h + 3, then the number of cycles of length k in a generalized de Bruijn digraph G B (n, d) is given by 1 /k Σ l\k μ(k/l)g l [d l / g l ], where μ is the Mobius function and [r] denotes the smallest integer not smaller than a real number r.

Minimum Degree Conditions and Optimal Graphs for Completely Independent Spanning Trees

Completely independent spanning trees \(T_1,T_2,\ldots ,T_k\) in a graph G are spanning trees in G such that for any pair of distinct vertices u and v, the k paths in the spanning trees between u and v mutually have no common edge and no common vertex except for u and v. The concept finds applications in fault-tolerant communication problems in a network. Recently, it was shown that Dirac’s condition for a graph to be hamiltonian is also a sufficient condition for a graph to have two completely independent spanning trees. In this paper, we generalize this result to three or more completely independent spanning trees. Namely, we show that for any graph G with \(n \ge 7\) vertices, if the minimum degree of a vertex in G is at least \(n-k\), where \(3 \le k \le \frac{n}{2}\), then there are \(\lfloor \frac{n}{k} \rfloor \) completely independent spanning trees in G. Besides, we improve the lower bound of \(\frac{n}{2}\) on the Dirac’s condition for completely independent spanning trees to \(\frac{n-1}{2}\) except for some specific graph. Our results are theoretical ones, since these minimum degree conditions can be applied only to a very dense graph. We then present constructions of symmetric regular graphs which include optimal graphs with respect to the number of completely independent spanning trees.