On Disjoint hypercubes in Fibonacci cubes

Sylvain Gravier (1) , Michel Mollard (1) , Simon Spacapan (2) , Sara Sabrina Zemljic (3)

1 IF - Institut Fourier
2 Faculty of Natural Sciences and Mathematics [Maribor]
3 Institute of Mathematics, Physics and Mechanics [Ljubljana]

The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma_n$, 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 $\Gamma_n$ 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 $\{q_{k}(n)\}_{n=0}^{ \infty}$.

Domaines: Combinatoire [math.CO]

Sylvain Gravier, Michel Mollard, Simon Spacapan, Sara Sabrina Zemljic. On Disjoint hypercubes in Fibonacci cubes. Discrete Applied Mathematics, 2015, 190-191, pp.50-55. ⟨hal-01139094⟩