Cube Qn Research Articles

Hamiltonian cycles and paths in hypercubes with disjoint faulty edges

We consider hypercubes with pairwise disjoint faulty edges. An n-dimensional hypercube Qn is an undirected graph with 2n nodes, each labeled with a distinct binary string of length n. The parity of the vertex is 0 if the number of ones in its label is even, and is 1 if the number of ones is odd. Two vertices a and b are connected by an edge iff a and b differ in one position. If a and b differ in position i, then we say that the edge (a,b) goes in dimension i and we define the parity of the edge as the parity of the end with 0 on the position i. It was already known that Qn is not Hamiltonian if all edges going in one dimension and of the same parity are faulty.In this paper we show that if n≥4 then all other hypercubes are Hamiltonian. In other words, every cube Qn, with n≥4 and disjoint faulty edges is Hamiltonian if and only if for each dimension there are two healthy crossing edges of different parity.

Circular embeddability of isometric words

Let f be a binary word and n≥1. Then the generalized Lucas cube Qn(f↽) is the graph obtained from the n-cube Qn by removing all vertices that have a circulation containing f as a factor. Ilić, Klavžar and Rho solved the question for which f and n, Qn(f↽) is an isometric subgraph of Qn for all binary words of length at most five. This question is further studied in this paper. For an isometric word f, sufficient and necessary conditions of Qn(f↽) being an isometric subgraph of Qn are found, and two problems on generalized Lucas cubes are also listed.

Daisy cubes and distance cube polynomial

Let X⊆{0,1}n. The daisy cube Qn(X) is introduced as the subgraph of Qn induced by the union of the intervals I(x,0n) over all x∈X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial DG,u(x,y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Qk at a given distance from the vertex u. It is proved that if G is a daisy cube, then DG,0n(x,y)=CG(x+y−1), where CG(x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then DG,u(x,−x)=1 holds for every vertex u in G.

Trimming and gluing Gray codes

We consider the algorithmic problem of generating each subset of [n]:={1,2,…,n} whose size is in some interval [k,l], 0≤k≤l≤n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k=0 and l=n this is the classical problem of generating all 2n subsets of [n] by element additions/removals, and for k=l this is the classical problem of generating all (nk) subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant O(1) time per generated set and O(n) space. Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Qn induced by all levels [k,l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q2k+1 induced by all levels [k−c,k+1+c], c∈{0,1,…,k}, has a Hamilton cycle. We also prove an approximate version of this generalized conjecture, showing that this graph has a cycle that visits a (1−o(1))-fraction of all vertices.

A negative answer to a problem on generalized Fibonacci cubes

Generalized Fibonacci cube Qn(f) is the graph obtained from the n-cube Qn by removing all vertices that contain a given binary string f as a consecutive substring. A binary string f is called bad if Qn(f) is not an isometric subgraph of Qn for some n, and the smallest such integer n, denoted by B(f), is called the index of f. Ilić, Klavžar and Rho posed a problem that if Qn(f) is not an isometric subgraph of Qn, is there a dimension n′ such that Qn(f) can be isometrically embedded into Qn′? We give a negative answer to this problem by showing that if f is bad, then for any n≥B(f), Qn(f) cannot be isometrically embedded to any hypercube.

On one problem for a simplex and a cube in ℝ n

Assume that S is a nondegenerate simplex in ℝn. Denote by α(S) the minimum σ > 0 such that the unit cube Qn:= [0, 1]n belongs to a translate of σS. In the case of σ(S) ≠ 1, a translate of α(S)S containing Qn is the image of S under homothety with the center at some point x ∈ ℝn. In this paper, we present the following computational formula for x. Denote by x(j) (j = 1, ..., n + 1) the vertices of S. Let A be an n + 1-by-n + 1 matrix such that its rows contain the coordinates of x(j); here, the last column of A consists of ones. Assume that A−1 = (lij). Then, the coordinates of x are the numbers $$x_k = \frac{{\sum\nolimits_{j = 1}^{n + 1} {\left( {\sum\nolimits_{i = 1}^n {\left| {l_{ij} } \right|} } \right)\left( {x_k^{(j)} - 1} \right)} }} {{\sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^{n + 1} {\left| {l_{ij} } \right| - 2} } }} (k = 1, \ldots ,n).$$ Since α(S) ≠ 1, the denominator in the right-hand side of this equality is distinct from zero. In addition, we present estimates for norms of projections under the linear interpolation of continuous functions given on Qn.

Hamiltonian cycles in hypercubes with [formula omitted] faulty edges

In this paper we consider Hamiltonian cycles in hypercubes with faulty edges and present a simple proof that the hypercube Qn is not Hamiltonian if it contains a subgraph T such that: (i) exactly half of the vertices in T have parity 0 (the other half have parity 1) and (ii) all edges joining the nodes of parity 0 (or 1) with nodes outside T, are faulty. We will call such a subgraph a trap disconnected halfway. Moreover, we show that for n⩾5, the cube Qn with 2n-4 faulty edges is not Hamiltonian only if it contains a trap T disconnected halfway with T being a subcube of dimension 1 or 2.

On Partitional Labelings of Graphs

The notion of partitional graphs, a subclass of sequential graphs, is introduced, and the cartesian product of a partitional graph and K2 is shown to be partitional. Every sequential graph is harmonious and felicitous. The partitional property of some bipartite graphs including the n-dimensional cube Qn is studied, and thus this paper extends what was known about the sequentialness, harmoniousness and felicitousness of such graphs.

New regular embeddings of n‐cubes Qn

AbstractWe investigate highly symmetrical embeddings of the n‐dimensional cube Qn into orientable compact surfaces which we call regular embeddings of Qn. We derive some general results and construct some new families of regular embeddings of Qn. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Qn which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004

Connectivity properties of random subgraphs of the cube

AbstractThe n‐dimensional cube Qn is the graph whose vertices are the subsets of {1, n} where two such vertices are adjacent if and only if their symmetric difference is a singleton. Clearly Qn is an n‐connected graph of diameter and radius n. Write M = n2n−1 = e(Qn) for the size of Qn. Let Q̃ = (Qt)M0 be a random Q̃‐process. Thus Qt is a spanning subgraph of Qn of size t, and Qt is obtained from Qt‐1 by the random addition of an edge of Qn not in Qt‐1. Let t(k) = τ(Q̃ δ ≧ k) be the hitting time of the property of having minimal degree at least k. It is shown in [5] that, almost surely, at time t(1) the graph Qt becomes connected and that in fact the diameter of Qt at this point is n + 1. Here we generalize this result by showing that, for any fixed k≧2, almost surely at time t(k) the graph Qt acquires the extremely strong property that any two of its vertices are connected by k internally vertex‐disjoint paths each of length at most n, except for possibly one, which may have length n + 1. In particular, the hitting time of k‐connectedness is almost surely t(k). © 1995 John Wiley & Sons, Inc.

On the diameter and radius of randon subgraphs of the cube

AbstractThe n‐dimensional cube Qn is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qn has diameter and radious n. Write M = n2n‐1 = e(Qn) for the size of Qn. Let Q = (Qt)oM be a random Qn‐process. Thus Qt is a spanning subgraph of Qn of size t, and Qt is obtained from Qt–1 by the random addition of an edge of Qn not in Qt–1, Let t(k) = τ(Q;δ⩾k) be the hitting time of the property of having minimal degree at least k. We show that the diameter dt = diam (Qt) of Qt in almost every Q̃ behaves as follows: dt starts infinite and is first finite at time t(1), it equals n + 1 for t(1) ⩽ t(2) and dt, = n for t ⩾ t(2). We also show that the radius of Qt, is first finite for t = t(1), when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Qt, for t = Ω(M). © 1994 John Wiley & Sons, Inc.

The Evolution of Random Subgraphs of the Cube

Abstract be a random Qn”‐process, that is let Q0 be the empty spanning subgraph of the cube Qn and, for 1 ⩽ t ⩽ M = nN/2 = n2n−1, let the graph Qt be obtained from Qt−1 by the random addition of an edge of Qn not present in Qt−1. When t is about N/2, a typical Qt undergoes a certain “phase transition'': the component structure changes in a sudden and surprising way. Let t = (1 + ϵ) N/2 where ϵ is independent of n. Then all the components of a typical Qt have o(N) vertices if ϵ < 0, while if ϵ > 0 then, as proved by Ajtai, Komlós, and Szemerédi, a typical Qt has a “giant” component with at least α(ϵ)N vertices, where α(ϵ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ⩽ (1 − n−η) N/2, then all components of a typical Qt have at most nβ(η) vertices, where β(η) > 0. More importantly, if 60(log n)3/n ⩽ ϵ = ϵn = o(1), then the largest component of a typical Qt has about 2ϵN vertices, while the second largest component has order O(nϵ−2). Loosely put, the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.

Recursive equivalence types and octahedra

AbstractLet the word “graph” be used in the sense of a countable, connected, simple graph with at least one vertex. We write Qn and Ocn for the graphs associated with the n-cube Qn and the n-octahedron Ocn respectively. In a previous paper (Dekker, 1981) we generalized Qn and Qn to a graph QN and a cube QN, for any nonzero recursive equivalence type N. In the present paper we do the same for Ocn and Ocn. We also examine the nature of the duality between QN and OcN, in case N is an infinite isol. There are c RETs, c denoting the cardinality of the continuum.

L p Norms of Certain Kernels on the N-Dimensional Torus

In this paper we study a class of kernels FR which generalize the Bochner-Riesz kernels on the N-dimensional torus. Our main result consists in upper estimates for the LP norms of FR as R tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces. 1. Throughout this paper we identify the N-dimensional torus TN (N > 2) with the N-dimensional cube QN = {x E R N: 1/2 0 we form the means FR* g(t) = f(R -m)g(m)exp(2'7imt) (1) m where t E TN, m ranges on the integer lattice ZN, and * denotes the convolution in TN. In several cases one is able to estimate the L 1( TN) norm of the kernel FR(t) = E f(R 1m)exp(2'srimt) (2) m and (if 0 E S and f(O) = 1) deduce convergence results for the means (1) (as R -x oc) when g belongs to certain classes of periodic functions (see e.g. [1]-[3], [7], [8], [12]). For instance, if f(x) = (1 -I X12)8 when lxl 1, then FR = K,A is the familiar Bochner-Riesz kernel. In this case K. I. Babenko (cf. [1]) has shown that IIKRIIl is of the same order as R(N-1)/2 as R tends to infinity (N > 2, 0 < 8 < (N 1)/2), while Stein proved the estimate IIKR Il log R if 8 = (N 1)/2 [9]. Recently Yudin [12] obtained the estimate lF = O(R (N1)/2) if f is the characteristic function of a closed balanced set C, whose boundary has finite upper Minkowski measure. The method of Yudin (which also yields estimates for the LP( TN) norms of FR) uses Jackson approximation theorem, which in tum involves estimates for the L2(R N) modulus of continuity of f. Such a method can be adapted to a more general situation, as we show in this paper. In ??2-4 we consider kernels of the type (2) associated with functionsf whose derivatives of certain orders are controlled by (not necessarily positive) Received by the editors May 13, 1980. 1980 Mathematics Subject Classification. Primary 42B99; Secondary 46E35.

Bounds on the Coarseness of the n-Cube

AbstractThe coarseness, c(G), of a graph G is the maximum number of edge disjoint nonplanar subgraphs contained in G For the n-dimensional cube Qn we obtain the inequalities