P-th Order Fibonacci Cubes: Structure & Recurrence

• P-th order Fibonacci cubes are higher-order analogues of classical Fibonacci cubes defined by forbidding runs of 1s or enforcing spacing constraints in binary strings.
• They arise from two main constructions—path-power graphs and forbidden-run models—each leading to distinct enumerative recurrences and combinatorial properties.
• These structures are studied through independent set encodings, yielding insights into Hamiltonian, median, and isometric subgraph properties within hypercubes.

P-th order Fibonacci cubes are higher-order analogues of the classical Fibonacci cube, the subgraph of the hypercube induced by binary strings with no consecutive 1s. The phrase, however, is not fully standardized. In one established convention it denotes graphs obtained from powers of paths and independent sets; in another it denotes subgraphs of $Q_n$ induced by strings avoiding a run $1^p$; and in a related but distinct family it denotes strings in which any two 1s are separated by at least $p$ zeros. These constructions coincide in the classical case but diverge for higher parameters, so the defining constraint is mathematically more informative than the name alone (Codara et al., 2014, Mollard, 22 Jul 2025, Mollard, 11 Feb 2025).

1. Terminology and competing conventions

The classical Fibonacci cube $\Gamma_n$ is the induced subgraph of $Q_n$ on binary strings of length $n$ with no consecutive 1s. In the general theory of generalized Fibonacci cubes, one fixes a binary string $f$ and defines $Q_d(f)$ by deleting from $Q_d$ all vertices that contain $f$ as a factor; in particular, $1^p$0, and the graphs $1^p$1 are exactly the higher-order Fibonacci cubes obtained by forbidding runs of $1^p$2 consecutive 1s (Wei et al., 2015, Mollard, 22 Jul 2025).

Three conventions are especially relevant.

Construction	Admissible strings	Classical case
$1^p$3	any two 1s are at distance greater than $1^p$4	$1^p$5
$1^p$6	no substring $1^p$7	$1^p$8
$1^p$9	at least $p$0 zeros between two 1s	$p$1

The first line is the path-power formulation of Codara and D’Antona. The second is the recent “$p$2-th order generalized Fibonacci cube” convention. The third is the family of Fibonacci $p$3-cubes. This suggests that the expression “P-th order Fibonacci cube” should always be read together with its defining rule, because the indexing of the classical case shifts from $p$4 to $p$5 depending on the convention (Codara et al., 2014, Mollard, 22 Jul 2025, Mollard, 11 Feb 2025).

2. Path powers, independent sets, and Hasse-diagram cubes

In the path-power framework, the basic graph is the $p$6-th power of a path $p$7, with vertex set $p$8 and edges

$p$9

An independent set of $\Gamma_n$0 is therefore a subset of vertices in which no two chosen indices differ by at most $\Gamma_n$1. Under the characteristic-vector encoding, vertices correspond to binary strings in which any two 1s are separated by at least $\Gamma_n$2 zeros. The Hasse diagram $\Gamma_n$3 of the poset of independent sets ordered by inclusion has these strings as vertices, and two vertices are adjacent precisely when one is obtained from the other by adding or removing a single admissible 1 (Codara et al., 2012, Codara et al., 2014).

For $\Gamma_n$4, this Hasse diagram is exactly the classical Fibonacci cube. For $\Gamma_n$5, Codara and D’Antona regard $\Gamma_n$6 as a natural generalization of Fibonacci cubes, and in this framework a “$\Gamma_n$7-th order Fibonacci cube” is naturally interpreted as $\Gamma_n$8. The extreme cases are also transparent: $\Gamma_n$9 yields the full hypercube $Q_n$0, because every subset is independent, while $Q_n$1 forces independent sets of size at most one and the Hasse diagram becomes a star centered at the empty set (Codara et al., 2014).

This formulation is structurally close to order ideals and distributive lattices. Because $Q_n$2 is an induced subgraph of a hypercube, it is bipartite; because it is a Hasse diagram of a poset of independent sets, it is connected, with the empty set as a canonical root. The parameter $Q_n$3 controls the exclusion radius along the underlying path rather than the maximum run length of 1s, which is why this family is distinct from the no-$Q_n$4 construction (Codara et al., 2012).

3. Forbidden runs, generalized Fibonacci cubes, and parameter shifts

A different convention defines the $Q_n$5-th order generalized Fibonacci cube $Q_n$6 as the subgraph

$Q_n$7

that is, the induced subgraph of $Q_n$8 on binary strings that do not contain $Q_n$9 as a contiguous substring. Equivalently, a vertex of $n$0 has at most $n$1 consecutive 1s. In this notation the classical Fibonacci cube is recovered as $n$2, and the case $n$3 is often called the Tribonacci cube (Mollard, 22 Jul 2025, Wei et al., 24 Jan 2026).

This model admits a precise combinatorial encoding. For a vertex $n$4 of Hamming weight $n$5, the string $n$6 decomposes uniquely into blocks from the alphabet

$n$7

This yields a bijection between vertices of weight $n$8 and compositions of $n$9 into $f$0 parts in $f$1. The associated counts are expressed באמצעות $f$2-nomial coefficients, defined by

$f$3

Accordingly, the number of vertices of weight $f$4 in $f$5 is

$f$6

and the maximum possible weight is

$f$7

(Mollard, 22 Jul 2025).

A common source of confusion is the distinction between $f$8 and Fibonacci $f$9-cubes $Q_d(f)$0. In the latter, any two 1s are separated by at least $Q_d(f)$1 zeros, so $Q_d(f)$2. Thus $Q_d(f)$3 forbids long blocks of 1s, whereas $Q_d(f)$4 enforces long gaps of 0s between 1s (Mollard, 11 Feb 2025, Mollard, 22 Jul 2025). Wei and Yang sharpened this point further for two-parameter families $Q_d(f)$5 and $Q_d(f)$6, proving that

$Q_d(f)$7

In their terminology, the standard no-$Q_d(f)$8 higher-order Fibonacci cubes correspond to $Q_d(f)$9 (Wei et al., 2019).

4. Enumerative theory

The path-power and forbidden-run models have different enumerative mechanisms, but both are governed by Fibonacci-like recurrences.

For the path-power family $Q_d$0, let $Q_d$1 denote the number of independent $Q_d$2-subsets of $Q_d$3. Then

$Q_d$4

and the total number of vertices is

$Q_d$5

These numbers satisfy

$Q_d$6

Codara and D’Antona then define the $Q_d$7-Fibonacci sequence $Q_d$8 by

$Q_d$9

and prove the edge formula

$f$0

The edge count is therefore the self-convolution of the $f$1-Fibonacci sequence (Codara et al., 2014).

For the no-$f$2 family $f$3, the number of vertices is

$f$4

where the $f$5-th order generalized Fibonacci numbers satisfy

$f$6

The edge numbers satisfy the recurrence

$f$7

and their generating function is

$f$8

A later paper showed that for all $f$9, the size $1^p$00 has three parallel descriptions: an iteration form, a convolution form, and a linear form. The convolution formula is

$1^p$01

and the linear form expresses $1^p$02 as a linear combination of $1^p$03 consecutive $1^p$04-th order Fibonacci numbers with coefficients linear in $1^p$05 (Mollard, 22 Jul 2025, Wei et al., 24 Jan 2026).

These formulas show that higher-order Fibonacci cubes support two different but analogous enumerative paradigms: a binomial-sum and Hasse-diagram theory for path powers, and a $1^p$06-nomial and $1^p$07-step Fibonacci theory for forbidden runs.

5. Metric and structural properties

The most general ambient framework is the generalized Fibonacci cube $1^p$08, obtained by deleting from $1^p$09 all vertices containing a fixed factor $1^p$10. A string $1^p$11 is called good if $1^p$12 is an isometric subgraph of $1^p$13 for all $1^p$14, and bad otherwise. For a bad string, the index $1^p$15 is the smallest dimension in which isometry fails. The key metric results are that if there exist $1^p$16-critical words for $1^p$17, then $1^p$18 or $1^p$19; consequently every bad string satisfies

$1^p$20

and if $1^p$21 is an isometric subgraph of $1^p$22, then $1^p$23 is an isometric subgraph of $1^p$24 as well (Wei et al., 2015). Since higher-order no-run Fibonacci cubes are $1^p$25, these results apply directly to the pattern $1^p$26.

Within the specific family $1^p$27, the structural picture is more refined. These graphs are daisy cubes, hence partial cubes and isometric subgraphs of some hypercube. For $1^p$28, and more generally for $1^p$29, $1^p$30 is a median graph; for larger $1^p$31 and $1^p$32, it is not. The same source notes that $1^p$33-th order Fibonacci cubes are Hamiltonian and even bipancyclic for most parameters (Mollard, 22 Jul 2025).

For Fibonacci $1^p$34-cubes $1^p$35, the geometry is again different. They are bipartite, daisy cubes, and partial cubes, and their direction-wise edge counts admit the exact formula

$1^p$36

This leads to closed formulas for the cube polynomial, distance cube polynomial, Wiener index, Mostar index, and irregularity (Mollard, 11 Feb 2025).

In the two-parameter spacing models, product structure is highly rigid. O-Fibonacci $1^p$37-cubes are non-trivial Cartesian products if and only if $1^p$38 and $1^p$39; equivalently, the hypercube case is the only decomposable one. For $1^p$40, the corresponding $1^p$41-th order spacing cubes are Cartesian-prime (Wei et al., 2019).

6. Related families and the broader generalization landscape

Higher-order Fibonacci cubes sit in a larger ecosystem of Fibonacci-like graph families. Codara and D’Antona extend the path-power picture from paths to cycles, obtaining generalized Lucas cubes $1^p$42, an $1^p$43-Lucas sequence, and a mixed convolution formula in which the number of edges is expressed via $1^p$44 and $1^p$45. In particular, the cycle case yields

$1^p$46

together with a convolution identity mixing $1^p$47-Fibonacci and $1^p$48-Lucas sequences (Codara et al., 2014).

Another connection arises through maximal subcubes. For Fibonacci $1^p$49-cubes $1^p$50, the number $1^p$51 of maximal induced $1^p$52-cubes satisfies

$1^p$53

and the top vertices of these maximal cubes correspond to vertices of specific weight in the $1^p$54-th order generalized Fibonacci cube $1^p$55. This creates a direct bridge between the gap-based family $1^p$56 and the forbidden-run family $1^p$57 (Mollard, 22 Jul 2025).

The literature also contains higher-level generalizations along a different axis. Metallic cubes generalize Fibonacci cubes and Pell graphs by replacing the Fibonacci recurrence with

$1^p$58

while preserving induced-hypercube embeddings, canonical decompositions, medianity, and Hamiltonicity. Horadam cubes extend this further to the full second-order Horadam recurrence

$1^p$59

again retaining recursive decomposition, decomposition into grids, edge and degree formulas, cube polynomials, and Hamiltonian paths and cycles (Došlić et al., 2023, Podrug, 2024). These families are not $1^p$60-th order Fibonacci cubes in the strict no-$1^p$61 sense, but they show that the combinatorial techniques developed for Fibonacci cubes extend well beyond the classical binary model.

Taken together, these results place P-th order Fibonacci cubes at the intersection of three themes: forbidden-pattern subgraphs of hypercubes, independent-set lattices of graph powers, and recurrence-driven graph constructions. The terminology varies, but the central mathematical idea is stable: classical Fibonacci cubes admit several non-equivalent higher-order extensions, each with its own recurrence, convolution theory, and geometric behavior.