广义de Bruijn有向图的k-元控制集

G =(V,A)表示一个有向图,其中V和A4分别表示有向图G的点集和弧集.对集合Dk? V(G),如果对于任意点v ∈ V(G),都存在k个点ui,1≤i≤k(可能存在某个ui和v是同一点)使得(ui,v)∈ A(G),则称Dk是G的一个k-元控制集.有向图G的k-元控制数γxk(G)是G的最小k-元控制集所含点的数目...     

广义de Bruijn和Kautz有向图的双向控制集

设G=(V,A)是一个有向图,其中V和A分别表示有向图G的点集和弧集.对集合TV(G),如果对于任意点v∈V(G)\T,都存在点u,w∈T(u,w可能是同一点)使得(u,v),(v,w)∈A(G),则称T是G的一个双向控制集.有向图G的双向控制数γ~*(G)是G的最小双向控制集所含点的数目.提出了广义de Bruijn和Kautz有向图的双向控制数的新上界,改进了以前文献中提出的相关结论.此外,对某些特殊的广义de Bruijn和Kautz有向图,通过构造其双向控制集,进一步改进了它们双向控制数的上、下界.     

广义de Bruijn和Kautz有向图的距离控制数

对于任意的正整数(?),强连通图G的顶点子集D被称为距离(?)-控制集,是指对于任意顶点v(?)D,D中至少含有一个顶点u,使得距离dG(u,v)≤(?)．图G距离(?)- 控制数γe(G)是指G中所有距离(?)-控制集的基数的最小者．本文给出了广义de Bruijn 和广义Kautz有向图的距离(?)-控制数的上界和下界,并且给出当它们的距离2-控制数达到下界时的一个充分条件．从而得到对于de Bruijn有向图B(d,k)的距离2-控制数γ2(B(d,k))= ．在该文结尾,我们猜想Kautz有向图K(d,k)的距离2-控制数γ2(K(d,k))= .     

The directed genus of the de Bruijn graph

Define the directed genus, Γ(G), of an Eulerian digraph G to be the minimum value of p for which G has a 2-cell embedding in the orientable surface of genus p so that every face of the embedding is bounded by a directed circuit in G. The directed genus of the de Bruijn graph D_n is shown to be

     

Generalized de Bruijn Cycles

For a set of integers , we define a q-ary -cycle to be an assignment of the symbols 1 through q to the integers modulo q ⁿ so that every word appears on some translate of . This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss reduced cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of . We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of completely.AMS Subject Classification: 94A55, 05C70.     

Generalized de Bruijn graphs

Oriented graphs in which every pair of vertices can be connected by a unique path of given length (not depending on the choice of the pair of vertices) are studied. These graphs are a natural extension of the well-known de Bruijn graphs and retain their most important properties. Some results on the structure of and methods for constructing such graphs are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 540–548, October, 1997. Translated by O. V. Sipacheva     

二元de Bruijn网络的可靠性分析

证明了二元de Bruijn网络是极大限制边连通的,并且它们的最小限制边割只能分离一条孤立边或者一个三角形. 利用此结果分析了二元de Bruijn网络的可靠性,确定了它们的可靠多项式的前四项系数.     

Nicolaas Govert de Bruijn,the enchanter of friable integers

N.G. de Bruijn carried out fundamental work on integers having only small prime factors and the Dickman–de Bruijn function that arises on computing the density of those integers. In this he used his earlier work on linear functionals and differential–difference equations. We review his relevant work and also some later improvements by others.     

Perfect matchings of a graph associated with a binary de Bruijn digraph

Let B_n be the binary de Bruijn digraph of order n and W the quotient set of the set of vertices of B_n with respect to the equivalence relation of rotation. Let G be the graph which has W as the set of vertices and in which two elements C and H are adjacent when there exist a vertex v of C and a vertex u of H such that (v,u) is an arc of B_n. Recently the problem of establishing whether the graph G has a perfect matching was posed. In this paper we answer in the positive to this problem in the case of n odd.     

A note on certain de Bruijn sequences with forbidden subsequences

We consider words over a finite alphabet with certain uniqueness properties (a subsequence of length k does not occur more than once) and distance properties (at least j other symbols separate the occurrence of the same symbol). The maximal length of these words is realised by linear de Bruijn sequences with certain forbidden subsequences. We prove the existence of these maximal sequences.     

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.     

Error Correcting Sequence and Projective De Bruijn Graph

Let X be a finite set of q elements, and n, K, d be integers. A subset C ⊂ X ⁿ is an (n, K, d) error-correcting code, if #(C) = K and its minimum distance is d. We define an (n, K, d) error-correcting sequence over X as a periodic sequence {a _i } _i=0,1,... (a _i ∈ X) with period K, such that the set of all consecutive n-tuples of this sequence form an (n, K, d) error-correcting code over X. Under a moderate conjecture on the existence of some type of primitive polynomials, we prove that there is a error correcting sequence, such that its code-set is the q-ary Hamming code with 0 removed, for q > 2 being a prime power. For the case q = 2, under a similar conjecture, we prove that there is a error-correcting sequence, such that its code-set supplemented with 0 is the subset of the binary Hamming code [2 ^m  − 1, 2 ^m  − 1 − m, 3] obtained by requiring one specified coordinate being 0. Received: October 27, 2005. Final Version received: December 31, 2007     

Independence number of de Bruijn graphs

In this paper, we give several exact values of the independence number of a de Bruijn graph UB(d,D) and in the other cases, we establish pertinent lower and upper bounds of this parameter. We show that asymptotically, if d is even, the ratio of the number of vertices of a greatest independent set of UB(d,D) is .     

Minimum Eulerian circuits and minimum de Bruijn sequences

Given a digraph (directed graph) with a labeling on its arcs, we study the problem of finding the Eulerian circuit of lexicographically minimum label. We prove that this problem is NP-complete in general, but if the labelling is locally injective (arcs going out from each vertex have different labels), we prove that it is solvable in linear time by giving an algorithm that constructs this circuit. When this algorithm is applied to a de Bruijn graph, it obtains the de Bruijn sequences with lexicographically minimum label.     

HAMILTONIAN CYCLES AND DECOMPOSITIONS OFCAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

In this paper, Hamiltonian cycles and decompositions of Cayley digraphs are investigat-ed. Sufficient conditions are given for these two problems respectively. Furthermore, the conditions are also necesaary for 2-regular Cayley disraphs, In addition, some known results about theCartesian products of two directed cycles are also deduced.     

广义de Bruijn有向图及其叠线图的支撑树与欧拉环游的计数

给出了一类特殊的广义deBruijn有向图的支撑树与欧环游的数目的简洁表示式，并得到了广义deBruijn有向叠线图的支撑树与欧拉环境数目的计算公式。