Maximal hypercubes in Fibonacci and Lucas cubes

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Maximal hypercubes in Fibonacci and Lucas cubes

Michel Mollard

To cite this version:

Michel Mollard. Maximal hypercubes in Fibonacci and Lucas cubes. 2012. �hal-00657456�

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Maximal hypercubes in Fibonacci and Lucas cubes

Michel Mollard ^∗

CNRS Universit´e Joseph Fourier Institut Fourier, BP 74

100 rue des Maths, 38402 St Martin d’H`eres Cedex, France e-mail: michel.mollard@ujf-grenoble.fr

Abstract

The Fibonacci cube Γnis the subgraph of the hypercube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λnis obtained

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from Γn by removing vertices that start and end with 1. We characterize maximal induced hypercubes in Γn and Λn and deduce for anyp≤n the number of maximalp-dimensional hypercubes in these graphs.

Key words: hypercubes; cube polynomials; Fibonacci cubes; Lucas cubes;

AMS subject classifications: 05C31, 05A15, 26C10

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1 Introduction

An interconnection topology can be represented by a graph G = (V, E), where V denotes the processors and E the communication links. The distance d_G(u, v) between two vertices u, v of a graphGis the length of a shortest path connectingu andv. Anisometric subgraphH of a graphGis an induced subgraph such that for

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any vertices u, v ofH we have d_H(u, v) =d_G(u, v).

The hypercube of dimension n is the graph Qn whose vertices are the binary strings of lengthnwhere two vertices are adjacent if they differ in exactly one coor- dinate. Theweight of a vertex, w(u), is the number of 1 in the string u. Notice that the graph distance between two vertices of Qn is equal to the Hamming distance

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of the strings, the number of coordinates they differ. The hypercube is a popular interconnection network because of its structural properties.

Fibonacci cubes and Lucas cubes were introduced in [4] and [11] as new interconnection networks. They are isometric subgraphs of Qn and have also recurrent structure.

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A Fibonacci string of length n is a binary string b1b2. . . bn with bibi+1 = 0 for 1 ≤ i < n. The Fibonacci cube Γn (n ≥ 1) is the subgraph of Qn induced by the

∗Partially supported by the ANR Project GraTel (Graphs for Telecommunications), ANR-blan- 09-blan-0373-01

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u u u

01 00 10

u u u

u u 001

000 010

100

101 _u

u u

u u u

u u 0001 0000 0010

0100 1000

1001 1010

0101 _u

u u

u u u

u u

Figure 1: Γ2 = Λ2, Γ3, Γ4 and Λ3, Λ4

z z z

z z

z z z

z z

00001 00000 00010

00100 01000

10000 01010

01001

10001 00101

10100

10101 10010

z z z

z z

z z z

z z

z z z

z z 000001

000000 000010

000100 001000

010000

100000 001001

001010

101001 101000 101010

100001 100010

010010

000101

010100

010101 010001

100100

100101

Figure 2: Γ5 and Γ6

z z z

z

z z z

z z

z z z

z z

z

Figure 3: Λ5 and Λ6

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Fibonacci strings of length n. For convenience we also consider the empty string and set Γ0 =K1. Call a Fibonacci stringb1b2. . . bnaLucas stringif b1bn6= 1. Then the Lucas cube Λn (n ≥1) is the subgraph of Qn induced by the Lucas strings of

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length n. We also set Λ0 =K1.

Since their introduction Γn and Λn have been also studied for their graph theory properties and found other applications, for example in chemistry (see the survey [7]). Recently different enumerative sequences of these graphs have been determined.

Among them: number of vertices of a given degree[10], number of vertices of a

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given eccentricity[3], number of pair of vertices at a given distance[8] or number of isometric subgraphs isomorphic so someQ_k[9]. The counting polynomial of this last sequence is known as cubic polynomial and has very nice properties[1].

We propose to study an other enumeration and characterization problem. For a given interconnection topology it is important to characterize maximal hypercubes,

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for example from the point of view of embeddings. So let us consider maximal hypercubes of dimension p, i.e. induced subgraphs H of Γn (respectively Λn) that are isomorphic to Q_p, and such that there exists no induced subgraph H^′ of Γ_n (respectively Λn), H⊂H^′, isomorphic to Qp+1.

Let fn,p and gn,p be the numbers of maximal hypercubes of dimension p of Γn,

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respectively Λ_n, and C^′(Γ_n, x) = P∞

p=0f_n,px^p, respectively C^′(Λ_n, x)P∞

p=0g_n,px^p, their counting polynomials.

By direct inspection, see figures 1 to 3, we obtain the first of them:

C^′(Γ0, x) = 1 C^′(Λ0, x) = 1 C^′(Γ1, x) = x C^′(Λ1, x) = 1 C^′(Γ2, x) = 2x C^′(Λ2, x) = 2x C^′(Γ3, x) = x²+x C^′(Λ3, x) = 3x C^′(Γ4, x) = 3x² C^′(Λ4, x) = 2x² C^′(Γ5, x) = x³+ 3x² C^′(Λ5, x) = 5x² C^′(Γ6, x) = 4x³+x² C^′(Λ6, x) = 2x³+ 3x²

The intersection graph of maximal hypercubes (also called cube graph) in a graph have been studied by various authors, for example in the context of median

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graphs[2]. Hypercubes playing a role similar to cliques in clique graph. Nice result have been obtained on cube graph of median graphs, and it is thus of interest, from the graph theory point of view, to characterize maximal hypercubes in families of graphs and thus obtain non trivial examples of such graphs. We will first characterize maximal induced hypercubes in Γnand Λnand then deduce the number of maximal

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p-dimensional hypercubes in these graphs.

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2 Main results

For any vertexx=x1. . . xn ofQn and anyi∈ {1, . . . , n} letx+ǫi be the vertex of Q_n defined by (x+ǫ_i)_i= 1−x_i and (x+ǫ_i)_j =x_j forj6=i.

Let H be an induced subgraph of Qn isomorphic to some Qk. The support

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of H is the subset set of {1. . . n} defined by Sup(H) = {i/∃x, y ∈ V(H) with x_i 6= y_i}. Let i /∈ Sup(H), we will denote by H+ǫe _i the subgraph induced by V(H)∪ {x+ǫi/x∈V(H)}. Note that H+ǫe i is isomorphic to Qk+1.

The following result is well known[6].

Proposition 2.1 In every induced subgraph H of Qn isomorphic toQk there exists

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a unique vertex of minimal weight, the bottom vertex b(H). There exists also a unique vertex of maximal weight, the top vertex t(H). Furthermore b(H) and t(H) are at distance k and characterizeH among the subgraphs ofQn isomorphic to Qk. We can precise this result. A basic property of hypercubes is that ifx, x+ǫi, x+ǫj

are vertices ofH thenx+ǫ_i+ǫ_j must be a vertex of H. By connectivity we deduce

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that ifx, x+ǫi and y are vertices ofH theny+ǫi must be also a vertex of H. We have thus by induction on k:

Proposition 2.2 if H is an induced subgraph ofQn isomorphic to Qk then (i) |Sup(H)|=k

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(ii) If i /∈Sup(H) then ∀x∈V(H) xi =b(H)i =t(H)i

(iii) If i∈Sup(H) then b(H)i= 0 andt(H)i = 1 (iv) V(H) ={x=x1. . . x_n/∀i /∈Sup(H) x_i=b(H)_i}.

If H is an induced subgraph of Γ_n, or Λ_n, then, as a set of strings of length n, it defines also an induced subgraph ofQn; thus Propositions 2.1 and 2.2 are still true

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for induced subgraphs of Fibonacci or Lucas cubes.

A Fibonacci string can be view as blocks of 0’s separated by isolated 1’s, or as isolated 0’s possibly separated by isolated 1’s. These two points of view give the two following decompositions of the vertices of Γn.

Proposition 2.3 Any vertex of weight w from Γn can be uniquely decomposed as

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0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where p=w; Pp

i=0l_i=n−w; l0, l_p ≥0 and l1, . . . , l_p−1≥1.

Proposition 2.4 Any vertex of weight w from Γn can be uniquely decomposed as 1^k⁰01^k¹. . .01^kⁱ. . .01^k^q where q =n−w; Pq

i=0k_i=w andk0, . . . , k_q≤1.

Proof. A vertex from Γn, n≥ 2 being the concatenation of a string of V(Γn−1) with 0 or a string ofV(Γ_n−2) with 01, both properties are easily proved by induction

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on n.

Using the the second decomposition, the vertices of weight w from Γ_n are thus obtained by choosing, in {0,1, . . . , q}, the w values of i such that ki = 1 in . We have then the classical result:

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Proposition 2.5 For any w ≤ n the number of vertices of weight w in Γn is

n−w+1 w

.

Considering the constraint on the extremities of a Lucas string we obtain the two following decompositions of the vertices of Λn.

Proposition 2.6 Any vertex of weight w in Λ_n can be uniquely decomposed as

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0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where p = w, Pp

i=0li = n−w, l0, lp ≥ 0, l0 +lp ≥ 1 and l1, . . . , lp−1≥1.

Proposition 2.7 Any vertex of weight w in Λn can be uniquely decomposed as 1^k⁰01^k¹. . .01^kⁱ. . .01^k^q whereq=n−w;Pq

i=0k_i =w;k0+k_q ≤1andk0, . . . , k_q ≤1.

From Propositions 2.2 and 2.3 it is possible to characterize the bottom and top

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vertices of maximal hypercubes in Γn.

Lemma 2.8 If H is a maximal hypercube of dimension p in Γn then b(H) = 0ⁿ and t(H) = 0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where Pp

i=0li =n−p; 0≤l0≤1;0≤lp ≤1 and 1 ≤l_i ≤2 for i= 1, . . . , p−1. Furthermore any such vertex is the top vertex of a unique maximal hypercube.

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Proof. LetHbe a maximal hypercube in Γ_n. Assume there exists an integerisuch thatb(H)i = 1. Theni /∈sup(H) by Proposition 2.2. Therefore, for anyx ∈V(H), xi =b(H)i = 1 thus x+ǫi ∈V(Γn). Then H+ǫe i must be an induced subgraph of Γ_n, a contradiction withH maximal.

Consider now t(H) = 0^l⁰10^l¹. . .10^lⁱ. . .10^l^p.

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If l0 ≥2 then for any vertex x of H we have x0 =x1 = 0, thus x+ǫ0 ∈V(Γn).

Therefore H+ǫe _i is an induced subgraph of Γ_n, a contradiction with H maximal.

The caselp ≥2 is similar by symmetry.

Assume now l_i ≥3, for some i∈ {1, . . . , p−1}. Let j =i+Pi−1

k=0l_k. We have thust(H)_j = 1 andt(H)_j+1 =t(H)_j+2 =t(H)_j+3=0. Then for any vertex x of H

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we have xj+1 =xj+2 =xj+3 = 0, thus x+ǫj+2 ∈V(Γn) and H is not maximal, a contradiction.

Conversely consider a vertex z = 0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where Pp

i=0l_i =n−p;

0≤l0 ≤1; 0≤lp≤1 and 1≤li≤2 fori= 1, . . . , p−1. Then, by Propositions 2.1 and 2.2,t(H) =zandb(H) = 0ⁿ define a unique hypercubeH inQnisomorphic to

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Q_pand clearly all vertices ofHare Fibonacci strings. Notice that for anyi /∈Sup(H) z+ǫi is not a Fibonacci string thus H is maximal.

With the same arguments we obtain for Lucas cube:

Proposition 2.9 If H is a maximal hypercube of dimension p ≥ 1 in Λn then b(H) = 0ⁿ and t(H) = 0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where Pp

i=0li = n−p; 0 ≤ l0 ≤ 2;

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0 ≤ l_p ≤ 2; 1 ≤ l0+l_p ≤ 2 and 1 ≤ l_i ≤ 2 for i= 1, . . . , p−1. Furthermore any such vertex is the top vertex of a maximal hypercube.

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Theorem 2.10 Let 0 ≤ p ≤ n and fn,p be the number of maximal hypercubes of dimension p in Γn then:

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fn,p=

p+ 1 n−2p+ 1

Proof. This is clearly true for p= 0 so assume p≥1. Since maximal hypercubes of Γnare characterized by their top vertex, let us consider the setT of strings which can be write 0^l⁰10^l¹. . .10^lⁱ. . .10^l^p where Pp

i=0l_i = n−p; 0≤ l0 ≤ 1; 0 ≤ l_p ≤ 1

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and 1 ≤ li ≤ 2 for i = 1, . . . , p−1. Let l_i^′ = li −1 for i = 1, . . . , p−1; l^′0 = l0; l^′_p = lp. We have thus a 1 to 1 mapping between T and the set of strings D = {0^l^′⁰10^l^′¹. . .10^l^′ⁱ. . .10^l^′^p} wherePp

i=0l_i^′ =n−2p+ 1 any l^′_i≤1 fori= 0, . . . , p. This set is in bijection with the setE ={1^l^′⁰01^l^′¹. . .01^l^′ⁱ. . .01^l^′^p}. By Proposition 2.4,Eis the set of Fibonacci strings of lengthn−p+ 1 and weightn−2p+ 1 and we obtain

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the expression of f_n,p by Proposition 2.5.

Corollary 2.11 The counting polynomial C^′(Γ_n, x) = P∞

p=0f_n,px^p of the number of maximal hypercubes of dimension p in Γn satisfies:

C^′(Γ_n, x) = x(C^′(Γ_n−2, x) +C^′(Γ_n−3, x)) (n≥3) C^′(Γ0, x) = 1, C^′(Γ1, x) =x, C^′(Γ2, x) = 2x

The generating function of the sequence {C^′(Γn, x)} is:

X

n≥0

C^′(Γn, x)yⁿ = 1 +xy(1 +y) 1−xy²(1 +y)

Proof. By theorem 2.10 and Pascal identity we obtain f_n,p=f_n−2_,p−1+f_n−3_,p−1 for n ≥3 and p ≥ 1. Notice that fn,0 = 0 for n 6= 0. The recurrence relation for

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C^′(Γ_n, x) follows. Settingf(x, y) =P

n≥0C^′(Γ_n, x)yⁿwe deduce from the recurrence relation f(x, y)−1−xy−2xy² =x(y²(f(x, y)−1) +y³f(x, y)) thus the value of

f(x, y).

Theorem 2.12 Let 1 ≤ p ≤ n and g_n,p be the number of maximal hypercubes of dimension p in Λn then:

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gn,p= n p

p n−2p

.

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Proof. The proof is similar to the previous result with three cases according to the value ofl0:

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By Proposition 2.9 the setT of top vertices that begin with 1 is the set of strings which can be write 10^l¹. . .10^lⁱ. . .10^l^p where Pp

i=1l_i = n−p and 1 ≤ l_i ≤ 2 for i= 1, . . . , p. Letl^′_i=li−1 fori= 1, . . . , p. We have thus a 1 to 1 mapping between T and the set of stringsD={10^l¹^′ . . .10^l^′ⁱ. . .10^l^p^′}wherePp

i=1l^′_i =n−2p, 0≤l_i^′ ≤1 for i = 1, . . . , p. Removing the first 1, and by complement, this set is in bijection

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with the set E={1^l^′¹. . .01^l^′ⁱ. . .01^l^p^′}. By Proposition 2.4,E is the set of Fibonacci strings of length n−p−1 and weight n−2p. Thus|T|= _n−2^p _p

.

The set U of top vertices that begin with 01 is the set of strings which can be write 010^l¹. . .10^lⁱ. . .10^l^p wherePp

i=1li =n−p−1; 1≤li≤2 fori= 1, . . . , p−1 and lp ≤1. Letl^′_i=li−1 fori= 1, . . . , p−1 andl^′_p=lp. We have thus a 1 to 1 mapping

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betweenU and the set of strings F ={010^l¹^′ . . .10^l^′ⁱ. . .10^l^p^′}wherePp

i=1l_i^′ =n−2p and l_i^′ ≤1 for i= 1, . . . , p. Removing the first 01, and by complement, this set is in bijection with the setG ={1^l¹^′ . . .01^l^′ⁱ. . .01^l^′^p}. By Proposition 2.4, G is the set of Fibonacci strings of lengthn−p−1 and weight n−2p. Thus|U|= _n−^p₂_p

. The last set, V, of top vertices that begin with 001, is the set of strings which

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can be write 0010^l¹. . .10^lⁱ. . .0^l^p−¹1 where Pp−1

i=1 l_i = n−p−2 and 1 ≤l_i ≤2 for i= 1, . . . , p−1. Let l^′_i=li−1 for i= 1, . . . , p−1. We have thus a 1 to 1 mapping between V and the set of strings H = {0010^l^′¹. . .10^l^′ⁱ. . .0^l^′^p−¹1} where Pp−1

i=1 l^′_i = n−2p−1 andl^′_i ≤1 fori= 1, . . . , p−1. Removing the first 001 and the last 1, this set, again by complement, is in bijection with the set K = {1^l^′¹. . .01^l^′ⁱ. . .01^l^′^p−¹}.

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The setK is the set of Fibonacci strings of length n−p−3 and weight n−2p−1.

Thus|V|= _n−2^p−1_p−1

andgn,p= 2 _n−2^p _p

+ _n−2^p−1_p−1

= ⁿ_p _n−2^p _p

.

Corollary 2.13 The counting polynomial C^′(Λn, x) = P∞

p=0gn,px^p of the number of maximal hypercubes of dimension p in Λ_n satisfies:

C^′(Λn, x) = x(C^′(Λn−2, x) +C^′(Λn−3, x)) (n≥5)

C^′(Λ0, x) = 1, C^′(Λ1, x) = 1, C^′(Λ2, x) = 2x, C^′(Λ3, x) = 3x, C^′(Λ4, x) = 2x²

The generating function of the sequence {C^′(Λ_n, x)} is:

X

n≥0

C^′(Λ_n, x)yⁿ= 1 +y+xy²+xy³−xy⁴ 1−xy²(1 +y)

Proof. Assume n ≥ 5. Here also by theorem 2.12 and Pascal identity we get

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g_n,p=g_n−2_,p−1+g_n−3_,p−1 forn≥5 andp≥2. Notice that whenn≥5 this equality occurs also for p = 1 and gn,0 = 0. The recurrence relation for C^′(Λn, x) follows and g(x, y) = P

n≥0C^′(Λn, x)yⁿ satisfies g(x, y)−1−y−2xy² −3xy³−2x²y4 = x(y²(g(x, y)−1−y−2xy²) +y³(g(x, y)−1−y)).

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Notice that fn,p 6= 0 if and only if _n

3

≤p≤_n+1 2

and gn,p 6= 0 if and only if

190 _n

3

≤ p ≤_n

2

(for n 6= 1). Maximal induced hypercubes of maximum dimension are maximum induced hypercubes and we obtain again that cube polynomials of Γ_n, respectively Λ_n, are of degree _n+1

2

, respectively≤_n

2

[9].

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