31 research outputs found

    Edges in Fibonacci cubes, Lucas cubes and complements

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    The Fibonacci cube of dimension n, denoted as Γ_n\Gamma\_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\Gamma\_n it is proved that the irregularity of Γ_n\Gamma\_n and Λ_n\Lambda\_n are two times the number of edges of Γ_n1\Gamma\_{n-1} and 2n2n times the number of vertices of Γ_n4\Gamma\_{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 results. For these two graphs we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach we determine the number of edges and the sequence of degrees of the cube complement of Γ_n\Gamma\_n

    The degree sequence of Fibonacci and Lucas cubes

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    AbstractThe Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γn and Λn is ∑i=0k(n−2ik−i)(i+1n−k−i+1) and ∑i=0k[2(i2i+k−n)(n−2i−1k−i)+(i−12i+k−n)(n−2ik−i)], respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp. low degree in Γn is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γn and Λn are easily computed

    On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes

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    Abstract Let Γ n and Λ n be the n-dimensional Fibonacci cube and Lucas cube, respectively. The domination number γ of Fibonacci cubes and Lucas cubes is studied. In particular it is proved that γ(Λ n ) is bounded below by , where L n is the n-th Lucas number. The 2-packing number ρ of these cubes is also studied. It is proved that and the exact values of ρ(Γ n ) and ρ(Λ n ) are obtained for n ≤ 10. It is also shown that Aut(Γ n ) Z 2

    Generalized Fibonacci cubes

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    AbstractGeneralized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed

    Author index to volume 255

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    Master index to volumes 251-260

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    Algebraic Analysis of Vertex-Distinguishing Edge-Colorings

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    Vertex-distinguishing edge-colorings (vdec colorings) are a restriction of proper edge-colorings. These special colorings require that the sets of edge colors incident to every vertex be distinct. This is a relatively new field of study. We present a survey of known results concerning vdec colorings. We also define a new matrix which may be used to study vdec colorings, and examine its properties. We find several bounds on the eigenvalues of this matrix, as well as results concerning its determinant, and other properties. We finish by examining related topics and open problems
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