133 research outputs found
Non covered vertices in Fibonacci cubes by a maximum set of disjoint hypercubes
The Fibonacci cube of dimension n, denoted as 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 n are covered by a maximum
set of disjoint subgraphs isomorphic to Q k , answering an open problem
proposed in [2]
Maximal hypercubes in Fibonacci and Lucas cubes
The Fibonacci cube is the subgraph of the hypercube induced by the
binary strings that contain no two consecutive 1's. The Lucas cube
is obtained from by removing vertices that start and end with 1. We
characterize maximal induced hypercubes in and and
deduce for any the number of maximal -dimensional hypercubes in
these graphs
On perfect codes in Cartesian products of graphs
AbstractAssuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph G we give the construction of a perfect code in the Cartesian product Gâ–¡Gâ–¡P2. Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations
Edges in Fibonacci cubes, Lucas cubes and complements
The Fibonacci cube of dimension n, denoted as , 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 it is proved that the
irregularity of and are two times the number of edges
of and times the number of vertices of ,
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
On Disjoint hypercubes in Fibonacci cubes
The {\em Fibonacci cube} of dimension , denoted as , is the
subgraph of -cube induced by vertices with no consecutive 1's. We
study the maximum number of disjoint subgraphs in isomorphic to
, and denote this number by . We prove several recursive results
for , in particular we prove that . We also prove a closed formula in which is given in
terms of Fibonacci numbers, and finally we give the generating function for the
sequence
On the nonexistence of three-dimensional tiling in the Lee metric II
AbstractWe prove that there does not exist a tiling of R3 with Lee spheres of radius greater than 0 such that the radius of at least one of them is greater than one
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