August 30, 2020 12:9 112-IJFCS 2050031 International Journal of Foundations of Computer Science Vol. 31, No. 5 (2020) 639{661 c World Scientific Publishing Company DOI: 10.1142/S0129054120500318 k-Fibonacci Cubes: A Family of Subgraphs of Fibonacci Cubes Omer¨ E˘gecio˘glu∗ Department of Computer Science University of California Santa Barbara Santa Barbara, California 93106, USA [email protected] Elif Saygı Department of Mathematics and Science Education Hacettepe University, Ankara 06800, Turkey [email protected] Z¨ulf¨ukar Saygı Department of Mathematics TOBB University of Economics and Technology Ankara 06560, Turkey [email protected] Received 29 November 2019 Accepted 19 February 2020 Published 26 August 2020 Communicated by Oscar Ibarra Hypercubes and Fibonacci cubes are classical models for interconnection networks with interesting graph theoretic properties. We consider k-Fibonacci cubes, which we obtain as subgraphs of Fibonacci cubes by eliminating certain edges during the fundamental recursion phase of their construction. These graphs have the same number of vertices as Fibonacci cubes, but their edge sets are determined by a parameter k. We obtain properties of k-Fibonacci cubes including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain. Keywords: Hypercube; Fibonacci cube; Fibonacci number. 1. Introduction An interconnection network can be represented by a graph G = (V; E) with ver- tex set V denoting the processors and edge set E denoting the communication links between the processors. One of the basic models for these networks is the n- dimensional hypercube graph Qn, whose vertices are indexed by all binary strings ∗Corresponding author. 639 August 30, 2020 12:9 112-IJFCS 2050031 640 O.¨ E˘gecio˘glu,E. Saygı & Z. Saygı of length n and two vertices are adjacent if and only if their Hamming distance is 1. Hsu [4], defined the n-dimensional Fibonacci cubes Γn as an alternative model of computation for interconnection networks and showed that they have interest- ing properties. Γn is a subgraph of Qn, where the vertices are those without two consecutive 1's in their binary string representation. Γ0 is defined as Q0, the graph with a single vertex and no edges. Numerous graph theoretic properties of Fibonacci cubes have been studied. In [8], a survey of the some of the properties including representations, recursive con- struction, hamiltonicity, degree sequences and other enumeration results are given. The induced d-dimensional hypercubes Qd in Γn are studied in [3, 9, 12, 13, 16, 17] and the boundary enumerator polynomial of hypercubes in Γn is considered in [18]. The number of vertex and edge orbits of Fibonacci cubes are determined in [1]. A Fibonacci string of length n which indexes the vertices of Γn is a binary string b b : : : b such that b b = 0 for 1 i n 1. The additional requirement 1 2 n i · i+1 ≤ ≤ − b b = 0 for n 2 defines a subgraph of Γ called Lucas cube Λ , which was also 1 · n ≥ n n proposed as a model for interconnection networks [14]. Other classes of graphs have also been defined along the lines of Fibonacci cubes. For instance subgraphs of Qn where k consecutive 1's are forbidden are proposed in [5] and Fibonacci (p; r)-cubes are presented in [2]. The generalized Fibonacci cube Qn(f) is defined as the subgraph of Qn by removing all the vertices that contain some forbidden string f [6]. With this formulation one has Γn = Qn(11). In this paper, we consider a special subgraph of Γn which is obtained by elim- inating certain edges. These are called k-Fibonacci cubes (or k-Fibonacci graphs) k and denoted by Γn as they depend on a parameter k. The edge elimination which defines k-Fibonacci cubes is carried out at the step analogous to where the funda- mental recursion is used to construct Γn from the two previous cubes by link edges. k The eliminated edges in Γn then recursively propagate according to the resulting fundamental construction which now depend on the value of the parameter k. k We obtain properties of Γn including the number of edges, the average degree of a vertex, the degree sequence and the number of hypercubes they contain. 2. Preliminaries Fibonacci numbers are defined by the recursion fn = fn 1 + fn 2 for n 2, with − − ≥ f0 = 0 and f1 = 1. Similarly, the Lucas numbers Ln are defined by the recursion Ln = Ln 1 + Ln 2 for n 2, with L0 = 2 and L1 = 1. Any positive integer − − ≥ can be uniquely represented as a sum of non-consecutive Fibonacci numbers. This representation is usually called the Zeckendorf or canonical representation. Here we note that this representation corresponds to the Fibonacci strings. For some positive n integer i assume that 0 i fn+2 1. Then i can be written as i = bj fn j+2, ≤ ≤ − j=1 · − where b 0; 1 . This gives that the Zeckendorf representation of i is (b ; b ; : : : ; b ) j 2 f g P 1 2 n and it corresponds to the Fibonacci string b1b2 : : : bn. Note that here we assume the integer 0 has Zeckendorf representation (0; 0;:::; 0). As an example, for n = 4, August 30, 2020 12:9 112-IJFCS 2050031 February 18, 2020 22:15 WSPC/INSTRUCTION FILE k-Fibonacci˙Cubes˙Egecioglu˙Saygi˙Feb19˙2020˙revised˙revised k-Fibonacci Cubes: A Special Family of Subgraphs of Fibonacci Cubes 641 7 = 5 + 2 = 1 f + 0 f + 1 f + 0 f gives that the Zeckendorf representation of · 5 · 4 · 3 · 2 7 is (1; 0; 1; 0) and this corresponds to the Fibonacci string 1010. The distance betweenk-Fibonacci two vertices cubes: Au specialand v familyin a connected of subgraphs graph of FibonacciG is defined cubes 3 as the length of a shortest path between u and v in G. For Qn and Γn this distance dimensionalcoincides with hypercube. the Hamming Then its distance vertex set (dH and) which edge is set the can number be written of different as bits in the string representation of the vertices. Let Qn = (V (Qn);E(Qn)) be the n- V (Q ) = 0, 1 n = b b . b b 0, 1 , 1 i n dimensional hypercube.n { Then} its{ vertex1 2 setn | andi ∈ edge { set} can≤ be≤ written} as E(Qn) = u, v n u, v V (Qn) and dH (u, v) = 1 . V (Qn) ={{0; 1 } |= b1∈b2 : : : bn bi 0; 1 ; 1 } i n f g f j 2 f g ≤ ≤ g n n 1 Of course V (Qn) = 2 and E(Qn) = n2 − . Similarly, we can write the vertex | E(Q| n) = u; v| u; v| V (Qn) and dH (u; v) = 1 : set and the edge set of Γffn = (Vg(Γ j n),E2(Γn)) as g n n 1 Of course V (Q ) = 2 and E(Q ) = n2 − . Similarly, we can write the vertex j n j j n j set and theV (Γ edgen) = setb1 ofb2 Γ. .= bn (Vb(Γi );E0, 1(Γ, ))1 asi n with bi bi+1 = 0 { n | ∈n { } n ≤ ≤ · } EV(Γ(Γn)) = = bu,b v : : :u, b v b V (Γ0n;)1 and; d1H (u,i v)n =with 1 . b b = 0 n {{f 1 2} | n j∈i 2 f g ≤ ≤ } i · i+1 g Note that theE(Γ number) = u; of v verticesu; v ofV the(Γ Fibonacci) and d ( cubeu; v) =Γn 1is: fn+2. n ff g j 2 n H g The fundamental decomposition [8] of Γn can be described as follows: Γn can Note that the number of vertices of the Fibonacci cube Γ is f . be decomposed into the subgraphs induced by the vertices thatn startn+2 with 0 and 10 The fundamental decomposition [8] of Γn can be described as follows: Γn can respectively. The vertices that start with 0 constitute a graph isomorphic to Γn 1 be decomposed into the subgraphs induced by the vertices that start with 0 and− 10 and the vertices that start with 10 constitute a graph isomorphic to Γn 2. This respectively. The vertices that start with 0 constitute a graph isomorphic− to Γn 1 decomposition can be written symbolically as − and the vertices that start with 10 constitute a graph isomorphic to Γn 2. This − decomposition can be writtenΓn symbolically= 0Γn 1 + 10Γ as n 2 . − − Γn = 0Γn 1 + 10Γn 2: In this representation the vertices are labeled− with− the Fibonacci strings. In Figure 1In the this first representation, six Fibonacci cubes the vertices are presented. are labeled The with labeling the Fibonacci of the vertices strings. follows In Fig. the 1 fundamentalthe first six recursion. Fibonacci For cubesn are2, presented. Γn is built The from labeling Γn 1 and of the Γn vertices2 by identifying follows the ≥ − − Γnfundamental2 with theΓ recursion.n 2 in Γn For1 byn what2, Γwen canis built call fromlink edges. Γn 1 Theand labels Γn 2 by of Γ identifyingn 1 stay − − − ≥ − − − unchangedΓn 2 with in the Γn Γ.n The2 in labels Γn 1 ofby the what vertices we can in call Γn link2 (whichedges. are The circled labels in of Figure Γn 1 stay 1) − − − − − areunchanged shifted up in by Γnf.n The+1 in labels Γn. of the vertices in Γn 2 (which are circled in Fig. 1) are − shifted up by fn+1 in Γn. Fig. 1. Fibonacci cubes Γ0; Γ1;:::; Γ5. Fig. 1. Fibonacci cubes Γ0, Γ1,..., Γ5. Throughout the paper we use the Fibonacci string representation and integer representation of the vertices interchangeably. For example, the first 8 vertices of Γ5 August 30, 2020 12:9 112-IJFCS 2050031 642 O.¨ E˘gecio˘glu,E.