The {\em Fibonacci cube} of dimension $n$, denoted as $Γ\_n$, is the subgraph of $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. We study the maximum number of disjoint subgraphs in $Γ\_n$ isomorphic to $Q\_k$, and denote this number by $q\_k(n)$. We prove several recursive results for $q\_k(n)$, in particular we prove that
$q\_{k}(n) = q\_{k-1}(n-2) + q\_{k}(n-3)$. We also prove a closed formula in which $q\_k(n)$ is given in terms of Fibonacci numbers, and finally we give the generating function for the sequence $\{q\_{k}(n)\}\_{n=0}^{ \infty}$.
The Fibonacci cube $Γ_n$ is the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $Λ_n$ is obtained from $Γ_n$ by removing vertices that start and end with 1. We characterize maximal induced hypercubes in $Γ_n$ and $Λ_n$ and deduce for any $p\leq n$ the number of maximal $p$-dimensional hypercubes in these graphs.
The Fibonacci cube of dimension n, denoted as $Γ$ n , is the subgraph of n-cube Q n induced by vertices with no consecutive 1's. In this short note we prove that asymptotically all vertices of $Γ$ n are covered by a maximum set of disjoint subgraphs isomorphic to Q k , answering an open problem proposed in [2].
Among the classical models for interconnection networks are hypercubes and Fibonacci cubes. Fibonacci cubes are induced subgraphs of hypercubes obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s, counted by Fibonacci numbers. Another set of binary strings which are counted by Fibonacci numbers are those with a restriction on the runlengths. Induced subgraphs of the hypercube on the latter strings as vertices define Fibonacci-run graphs. They have the same number of vertices as Fibonacci cubes, but fewer edges and different connectivity properties…
The optimal cube factor of a graph, a special kind of component factor, is first introduced. Furthermore, the optimal cube factors of Fibonacci and matchable Lucas cubes are studied; and some results on the Padovan sequence and binomial coefficients are obtained.
It is proved that the asymptotic average eccentricity and the asymptotic average degree of Fibonacci cubes and Lucas cubes are $(5+\sqrt 5)/10$ and $(5-\sqrt 5)/5$, respectively. A new labeling of the leaves of Fibonacci trees is introduced and proved that the eccentricity of a vertex of a given Fibonacci cube is equal to the depth of the associated leaf in the corresponding Fibonacci tree. Hypercube density is also introduced and studied. The hypercube density of both Fibonacci cubes and Lucas cubes is shown to be $(1-1/\sqrt 5)/\log_2φ$, where $φ$ is the golden ratio, and the Cartesian produ…
The diameter of a graph is the maximum distance between pairs of vertices in the graph. A pair of vertices whose distance is equal to its diameter are called diametrically opposite vertices. The collection of shortest paths between diametrically opposite vertices are referred as diametral paths. In this work, we enumerate the number of diametral paths for Fibonacci cubes, Lucas cubes and Alternate Lucas cubes. We present bijective proofs that show that these numbers are related to alternating permutations and are enumerated by Euler numbers.
A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in this paper. Several enumerative polynomials, e.g.\ rank generating function, cube polynomials and degree sequence polynomials are obtained. Some of these results relate to Fibonacci sequence and Padovan sequence.
The Mostar index of a graph was defined by Došlić, Martinjak, Škrekovski, Tipurić Spužević and Zubac in the context of the study of the properties of chemical graphs. It measures how far a given graph is from being distance-balanced. In this paper, we determine the Mostar index of two well-known families of graphs: Fibonacci cubes and Lucas cubes.
Let $p ,r $ and $n $ be positive integers. Then the O-Fibonacci $(p,r)$-cube $OΓ^{(p,r)}_{n}$ is the subgraph of $Q_{n}$ induced on the binary words in which there is at least $p-1$ zeros between any two $1$s and there is at most $r$ consecutive $10^{p-1}$. These cubes include a wide range of cubes as their special cases, such as hypercubes, Fibonacci cubes, and postal networks. In this note it is proved that $OΓ^{(p,r)}_{n}$ is a non-trivial Cartesian product if and only if $p=1$ and $r\geq n\geq2$.
The distinguishing number of a graph G, denoted D(G), is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we show that the distinguishing number of p-th graph power of the n-dimensional hypercube is 2 whenever 2 < p < n-1. This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube, a variant of the hypercube, answering an open question.
This paper introduces a new variant of hypercubes, which we call Z-cubes. The n-dimensional Z-cube $H_n$ is obtained from two copies of the (n-1)-dimensional Z-cube $H_{n-1}$ by adding a special perfect matching between the vertices of these two copies of $H_{n-1}$. We prove that the n-dimensional Z-cubes $H_n$ has diameter $(1+o(1))n/\log_2 n$. This greatly improves on the previous known variants of hypercube of dimension n, whose diameters are all larger than n/3. Moreover, any hypercube variant of dimension $n$ is an n-regular graph on $2^n$ vertices, and hence has diameter greater than $n/…
In recent years there has been much interest in certain subcubes of hypercubes, namely Fibonacci cubes and Lucas cubes (and their generalized versions). In this article we consider online routing of linear permutations on these cubes. The model of routing we use regards edges as bi-directional, and we do not allow queues of length greater than one. Messages start out at different vertices, and in movements synchronized with a clock, move to an adjacent vertex or remain where they are, so that at the next stage there is still exactly one message per vertex. This is the routing model we defined …
The {\em Fibonacci cube} of dimension $n$, denoted as $Γ\_n$, is the subgraph of the $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect codes in $Γ\_n$ for $n\geq 4$. As an open problem the authors suggest to consider the existence of perfect codes in generalization of Fibonacci cubes. The most direct generalization is the family $Γ\_n(1^s)$ of subgraphs induced by strings without $1^s$ as a substring where $s\geq 2$ is a given integer. We prove the existence of a perfect code in $Γ\_n(1^s)$ for $n=2…
The Fibonacci cube of dimension n, denoted as $Γ\_n$, is the subgraph of the hypercube induced by vertices with no consecutive 1's. The irregularity of a graph G is the sum of |d(x)-d(y)| over all edges {x,y} of G. In two recent paper based on the recursive structure of $Γ\_n$ it is proved that the irregularity of $Γ\_n$ and $Λ\_n$ are two times the number of edges of $Γ\_{n-1}$ and $2n$ times the number of vertices of $Γ\_{n-4}$, respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind (Figure 2) we give a bijective proof of both result…
Fibonacci cubes are induced subgraphs of hypercube graphs obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s. This class of graphs has been studied extensively and generalized in many different directions. Induced subgraphs of the hypercube on binary strings with restricted runlengths as vertices define Fibonacci-run graphs. These graphs have the same number of vertices as Fibonacci cubes, but fewer edges and different graph theoretical properties. Basic properties of Fibonacci-run graphs are presented in a companion paper, while in this paper we…
The Fibonacci cube $Γ_n$ is obtained from the $n$-cube $Q_n$ by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube $Λ_n$ is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of the sizes of the vertex orbits of $Λ_n$ is $\{k \ge 1;\ k \divides n\} \cup\, \{k \ge 18;\ k \divides 2n\}$, the number of the vert…
The generalized Fibonacci cube $Q_d(f)$ is the subgraph of the $d$-cube $Q_d$ induced on the set of all strings of length $d$ that do not contain $f$ as a substring. It is proved that if $Q_d(f) \cong Q_d(f')$ then $|f|=|f'|$. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. It is also proved that there exist pairs of strings $f, f'$ such that $Q_d(f) \cong Q_d(f')$, where $|f| \ge \frac{2}{3}(d+1)$ and $f'$ cannot be obtained from $f$ by its reversal or binary complementation. Strings $f$ and $f'$ with $|f|…
We study the prefixes of the Fibonacci word that end with a cube. Using Walnut we obtain an exact description of the positions of the Fibonacci word at which a cube ends.
In this paper, first it is shown that the "FSibonacci $(p,r)$-cube"(denoted as $IΓ_{n}^{(p,r)}$) studied in many papers, such as \cite{OZY}, \cite{K1}, \cite{OZ}, \cite{KR} and \cite{JZ}, is a new topological structure different from the original one (denoted as $OΓ_{n}^{(p,r)}$) presented by Egiazarian and Astola $\cite{EA}$. Then some topological properties of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$ are studied, including the recursive structure of them, the cubes $OΓ_{n}^{(p,r)}$ which are partial cubes and median graphs, some distance invariants of $IΓ_{n}^{(p,r)}$ and $OΓ_{n}^{(p,r)}$, and …
We introduce a family of nonnegative integer vectors - primitive vectors - defining hyperplanes of the real affine cube over $C^n:=\{-1,1\}^n$ and study their properties with respect to the rectangles of the cube. As a consequence we give a short proof that, for small dimensions ($n\leq 7$), the real affine cube can be recovered from its signed rectangles and its signed cocircuits complementary of its facets and skew-facets.
This paper considers a kind of generalized measure $λ_s^{(h)}$ of fault tolerance in a hypercube-like graph $G_n$ which contain several well-known interconnection networks such as hypercubes, varietal hypercubes, twisted cubes, crossed cubes and Möbius cubes, and proves $λ_s^{(h)}(G_n)= 2^h(n-h)$ for any $h$ with $0\leqslant h\leqslant n-1$ by the induction on $n$ and a new technique. This result shows that at least $2^h(n-h)$ edges of $G_n$ have to be removed to get a disconnected graph that contains no vertices of degree less than $h$. Compared with previous results, this result enhances fau…
The hypercube of dimension n is the graph whose vertices are the 2^n binary words of length n, and there is an edge between two of them if they have Hamming distance 1. We consider an edit distance based on swaps and mismatches, to which we refer as tilde-distance, and define the tilde-hypercube with edges linking words at tilde-distance 1. Then, we introduce and study some isometric subgraphs of the tilde-hypercube obtained by using special words called tilde-isometric words. The subgraphs keep only the vertices that avoid a given tilde-isometric word as a factor. In the case of word 11, the …
The connectivity is an important parameter to evaluate the robustness of a network. As a generalization, structure connectivity and substructure connectivity of graphs were proposed. For connected graphs $G$ and $H$, the $H$-structure connectivity $κ(G; H)$ (resp. $H$-substructure connectivity $κ^{s}(G; H)$) of $G$ is the minimum cardinality of a set of subgraphs $F$ of $G$ that each is isomorphic to $H$ (resp. to a connected subgraph of $H$) so that $G-F$ is disconnected or the singleton. As popular variants of hypercubes, the $n$-dimensional folded hypercubes $FQ_{n}$ and augmented cubes $AQ…
For a set $X$ of binary words of length $h$ the daisy cube $Q_h(X)$ is defined as the subgraph of the hypercube $Q_h$ induced by the set of all vertices on shortest paths that connect vertices of $X$ with the vertex $0 ^h$. A vertex in the intersection of all of these paths is a minimal vertex of a daisy cube. A graph $G$ isomorphic to a daisy cube admits several isometric embeddings into a hypercube. We show that an isometric embedding is proper if and only if the label $0 ^h$ is assigned to a minimal vertex of $G$. This result allows us to devise an algorithm which finds a proper embedding o…
Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets
Faculty of Mechanical Engineering, University of Ljubljana（卢布尔雅那大学机械工程学院）；Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那大学数学物理力学研究所）；Department of Mathematics, Penn State University Beaver Campus（宾夕法尼亚州立大学比弗分校数学系）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Faculty of Chemistry and Chemical Engineering, University of Maribor（马里博尔大学化学与化学工程学院）
A binary string $f$ is a factor of string $u$ if $f$ appears as a sequence of $|f|$ consecutive bits of $u$, where $|f|$ denotes the length of $f$. Generalized Fibonacci cube $Q_{d}(f)$ is the graph obtained from the $d$-cube $Q_{d}$ by removing all vertices that contain a given binary string $f$ as a factor. A binary string $f$ is called good if $Q_{d}(f)$ is an isometric subgraph of $Q_{d}$ for all $d\geq1$, it is called bad otherwise. The index of a binary string $f$, denoted by $B(f)$, is the smallest integer $d$ such that $Q_{d}(f)$ is not an isometric subgraph of $Q_{d}$. Ilić, Klavžar a…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of $n$-dimensional hypercube proposed by Erdős and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.
The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent sets of the h-th power of a path ordered by inclusion. For h=1 such a diagram is called a Fibonacci cube, and for h>1 we obtain a generalization of the Fibonacci cube. Consequently, we derive a generalized notion of Fibonacci sequence, called h-Fibonacci sequence. Then, we show that the number of edges of a generalized Fibonacci cube is obtained by con…
We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.
The size of $k$-th order generalized Fibonacci cubes
School of Mathematics and Statistics Science, Ludong University（鲁东大学数学与统计科学学院）；School of Mathematics and Information Science, Yantai University（烟台大学数学与信息科学学院）
As a generalization of the traditional connectivity, the g-component edge connectivity cλg(G) of a non-complete graph G is the minimum number of edges to be deleted from the graph G such that the resulting graph has at least g components. Hypercube-like networks (HL-networks for short) are obtained by manipulating some pairs of edges in hypercubes, which contain several famous interconnection networks such as twisted cubes, Mobius cubes, crossed cubes, locally twisted cubes. In this paper, we determine the (g + 1)-component edge connectivity of the n-dimensional HL-networks.
We find the asymptotic behavior of the Steiner k-diameter of the $n$-cube if $k$ is large. Our main contribution is the lower bound, which utilizes the probabilistic method.
The Fibonacci cube of dimension n, denoted as $Γ$ n , is the subgraph of the n-cube 5 Q n induced by vertices with no consecutive 1's. Ashrafi and his co-authors proved the non-existence of perfect codes in $Γ$ n for n $\ge$ 4. As an open problem the authors suggest to consider the existence of perfect codes in generalizations of Fibonacci cubes. The most direct generalization is the family $Γ$ n (1 s) of subgraphs induced by strings without 1 s as a substring where s $\ge$ 2 is a given integer. In a precedent work 10 we proved the existence of a perfect code in $Γ$ n (1 s) for n = 2 p -- 1 an…
The varietal hypercube $VQ_n$ is a variant of the hypercube $Q_n$ and has better properties than $Q_n$ with the same number of edges and vertices. This paper proves that $VQ_n$ is vertex-transitive. This property shows that when $VQ_n$ is used to model an interconnection network, it is high symmetrical and obviously superior to other variants of the hypercube such as the crossed cube.
It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $Θ(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $Θ(n^2\ln n)$. Similar estimations are obtained for systems of mutually strong orthogonal latin $d$-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order $n$ such that the logarithm of its cardinality is $Θ(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
A coloring of the $q$-ary $n$-dimensional cube (hypercube) is called perfect if, for every $n$-tuple $x$, the collection of the colors of the neighbors of $x$ depends only on the color of $x$. A Boolean-valued function is called correlation-immune of degree $n-m$ if it takes the value 1 the same number of times for each $m$-dimensional face of the hypercube. Let $f=χ^S$ be a characteristic function of some subset $S$ of hypercube. In the present paper it is proven the inequality $ρ(S)q({\rm cor}(f)+1)\leq α(S)$, where ${\rm cor}(f)$ is the maximum degree of the correlation immunity of $f$, $α(…
Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes $Q_n$. They are induced with intervals between chosen vertices of $Q_n$ and the vertex $0^n\in V(Q_n)$. In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klavžar and Mollard, 2018, in the introductory paper of daisy cubes \cite{KlaMol-18}. To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph $G$ isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we pre…
Let T1, T2,.... Tk be spanning trees in a graph G. If for any pair of vertices u and v of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges then T1, T2,.... Tk are called edge-disjoint spanning trees in G. The design of multiple edge-disjoint spanning trees has applications to the reliable communication protocols. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. For AQn (n > 2), construction of n-1 edge-disjoint spanning trees is given the result is optimal with respect…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. In this paper, we give the exact values of crossing numbers for some variations of hypercube with order at most four, including crossed cube, locally twisted cube and Möbius cube.
A paratopism is an action on a Latin hypercube of dimension d and order n which is an element of the wreath product $S_n \wr S_{d+1}$. A paratopism is said to be an autoparatopism if there is at least one Latin hypercube which is mapped to itself under the action of the paratopism. In this paper we classify autoparatopisms of Latin cubes given d = 3 and $n \in \mathbb{Z^+}$, upto the conjugacy in $S_n \wr S_4$. In order to achieve this objective, we prove that given an autoparatopism $σ \in S_n \wr S_{d+1}$, all the conjugates of $σ$ are autoparatpisms. Also, an important condition is presente…
Let $Q_d$ be the hypercube of dimension $d$ and let $H$ and $K$ be subsets of the vertex set $V(Q_d)$, called configurations in $Q_d$. We say that $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ onto $K$. Let $n\geq d$ be an integer, let $H$ be a configuration in $Q_d$ and let $S$ be a configuration in $Q_n$. We let $λ(H,d,n)$ be the maximum, over all configurations $S$ in $Q_n$, of the fraction of sub-$d$-cubes $R$ of $Q_n$ in which $S\cap R$ is an exact copy of $H$, and we define the $d$-cube density $λ(H,d)$ of $H$ to be the limit as $n$ goes to infin…
The hypercube Qn of dimension n is one of the most versatile and powerful interconnection networks. The n-dimensional folded cube denoted as FQn, a variation of the hypercube possesses some embeddable properties that the hypercube does not possess. Dong and Wang(In Theor. Comput. Sci.771(2019)93-98) conjectured that "A subset Em of edges of FQn is a perfect matching if and only if FQn - Em is isomorphic to Qn". In this paper, we disprove this conjecture by providing some perfect matchings removal of which from FQn do not give a graph isomorphic to Qn.
