Independent subsets of powers of paths, and Fibonacci cubes

Abstract

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.Comment: Preprint submitted to Electronic Notes in Discrete Mathematic

Generalized Fibonacci and Lucas cubes arising from powers of paths and cycles

“…Thus, there are n + 1 independent sets of P n . Notice that, as stated in the introduction, for each n, H (1) n is the Fibonacci cube Γ n .…”

“…Thus, for instance, P (0) n and C (0) n are the graphs made of n isolated nodes, P (1) n is the path with n vertices, and C (1) n is the cycle with n vertices. Figure 1 shows some powers of paths and cycles.…”

“…To the best of our knowledge, our H In the first part of this paper (which is an extended version of [1] -see remark at the end of this section) we evaluate p n . We then introduce a generalization of the Fibonacci sequence, that we call h-Fibonacci sequence and denote by F (h) .…”

“…Such integer sequence is based on the values of c Remark. As mentioned before, this work is an extended version of [1]. (The extended abstract [1] has been presented at the conference Combinatorics 2012, Perugia, Italy, 2012, and consequently published in ENDM.)…”

“…Specifically, Section 2 of the present paper is [1, Section 2] enriched with some details and full proofs, while Section 3 is [1, Section 3], with corrected notation and some remarks added. Lemma 3.1, Lemma 3.2, and Theorem 3.4 have remained as they were in [1]: we decided to keep full proofs of these results, to make the content clearer and in order for the paper to be self-contained.…”