Hamilton Cycle Decomposition of the Butterfly Network

Texte intégral

(1)Hamilton Cycle Decomposition of the Butterfly Network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Pérennes. To cite this version: Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Stéphane Pérennes. Hamilton Cycle Decomposition of the Butterfly Network. RR-2920, INRIA. 1996. �inria-00073777�. HAL Id: inria-00073777 https://hal.inria.fr/inria-00073777 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Hamilton cycle decomposition of the Butterfly network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Ste´phane Perennes. N˚ 2920 Juin 1996 THE`ME 1. ISSN 0249-6399. apport de recherche.

(3)

(4) Hamilton cycle decomposition of the Butterfly network Jean-Claude Bermond, Eric Darrot, Olivier Delmas, Ste´phane Perennes The`me 1 — Re´seaux et syste`mes Projet SLOOP Rapport de recherche n˚2920 — Juin 1996 — 16 pages.

(5) . . of degree and Abstract: in this paper, we prove that the wrapped Butterfly graph dimension is decomposable into Hamilton cycles. This answers a conjecture of D. Barth and A. Raspaud who solved the case .. . Key-words: Butterfly graph, graph theory, Hamiltonism, Hamilton decomposition, Hamilton cycle, Hamilton circuit, perfect matching.. (Re´sume´ : tsvp). This work has been supported by the CEFIPRA (French-Indian collaboration) and the European project HCM MAP. Version of June 28, 1996 to appear in PARALLEL P ROCESSING L ETTERS. Email : {bermond, darrot, delmas, sp}@unice.fr. Unite´ de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Te´le´phone : (33) 93 65 77 77 – Te´lećopie : (33) 93 65 77 65.

(6) Dećomposition en cycles Hamiltoniens du re´seau Butterfly.

(7) . . Re´sume´ : dans cet article, nous prouvons que le graphe Butterfly reboucle´ de degre´ et de dimension est dećomposable en cycles Hamiltoniens. Ce re´sultat re´pond a` une conjecture de D. Barth et A. Raspaud qui ont re´solu le cas .. . Mots-cle´ : graphe Butterfly, theórie des graphes, Hamiltonisme, dećomposition Hamiltonienne, cycle Hamiltonien, circuit Hamiltonien, couplage parfait..

(8) Hamilton cycle decomposition of the Butterfly network. 3. 1 Introduction and notations The construction of one, and if possible many edge-disjoint Hamilton cycles in a network can provide advantage for algorithms that make use of a ring structure. As example, the existence of many edge-disjoint Hamilton cycles allows the message traffic to be evenly distributed across the network. Furthermore, a partition of the edges into Hamilton cycles can be used in various distributed algorithms (termination, garbage collector, ). So, many authors have considered the problem of finding how many edge-disjoint Hamilton cycles can be found in a given network. The most significant results have been obtained for the class of Cayley graphs on abelian groups, and for (underlying) line digraphs. Here we solve this problem for the Butterfly networks. These networks have been proposed as suitable topologies for parallel computers, due to their interesting structure (see [8, 9]) because they are, when properly defined, both Cayley digraphs (on a non-abelian group) and iterated line digraphs.. 1.1 Definitions First, we have to warn the reader that under the name Butterfly and with the same notation, different networks are described. Indeed, if some authors consider the Butterfly network as a multistage network used to route permutations, others consider it as point-to-point network. In what follows, we will study the point-to-point version, and use Leigthon’s terminology [8], namely, wrapped Butterfly. Also, when we use the terms edge-disjoint or arc-disjoint, it obviously means pairwise edge-disjoint or arc-disjoint. In this article, we will use the next definitions and notations. For definitions not given here, see [9]. . will denote the set of integers modulo ; addition of elements in. . . . will always occur in. . .. . Definition 1.1 The wrapped Butterfly digraph of degree and dimension , denoted , .

(9) . where is an element of , that is, a word has as vertices the couples . ( is called the level). For any , a vertex where the letters belong to , and . ! #"%$ ! & #"(' $ is joined by an arc to vertices where is $ any element of . Each one of these arcs is said to have the slope .. . . . is a -regular digraph with . . . *)+'. vertices; its diameter is . This network is sometimes considered as undirected, but its structure being indeed directed, we will always consider the digraph. For convenience, we repeat the level , when drawing the wrapped Butterfly digraph. Hence, the reader has Figure (1) dis to.- remember that the two occurrences of level , have to be identified. plays with the arcs directed from left to right. Note that is often represented (for example in [8, 9]) in an opposite way to our drawing as the authors denote the nodes . .

(10) .. . . RR n˚2920. .

(11) 4. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. Levels 0. 1. Duplicated level 0. 00 01. Lines or words. 02. Vertex (10,0) 10 11 12. 20 21 22. Figure 1: The digraph. .- , the arcs beings directed from left to right.. Now, we give two digraph definitions we use in the following.. . . . will denote the complete symmetric digraph with a loop on each vertex,. . . . will denote the complete bipartite digraph where each set of the bipartition has size and with all the arcs directed from the left part to the right part. Note that. ' is nothing else than . . . .. In digraphs, the concept of dipaths and circuits (directed cycles) is well-known. Here, we need to use more general concepts valid for digraphs of paths and cycles (which are also called oriented elementary paths and oriented elementary cycles).. . . #

(12)

(13)

(14) where the and where the

(15). Definition 1.2 A path of a digraph is a sequence ’s are vertices and the ’s are arcs such that the end vertices of are and # . sequence does not meet twice the same vertex except maybe and. . Definition 1.3 A path such that. .

(16) . . in the sequence. . . is called a cycle.. . . or from to . If all the arcs of the Note that the arc can be either directed from to we have a dipath (resp. circuit also called dicycle). path (resp. cycle) are directed from to. . . . ". . . ). Definition 1.4 A vertex of a cycle is said to be of type (resp. of type ) for the cycle, if the terminal vertex of (resp. ) and the initial vertex of (resp. ).. is. INRIA.

(17) Hamilton cycle decomposition of the Butterfly network. Note that in a circuit, all vertices are of type vertices of a cycle.. . ". 5. , but the type is not necessarily defined for all the. . . . Definition 1.5 A vertex is said to be crossed by a cycle, or a cycle crosses the vertex , if is " ) of type or of type " for the cycle. When a vertex is crossed by a cycle, we will define its sign ' ) ' " ) function by (resp. ) if is of type (resp. of type ).. . . . . . . . . .

(18)

(19) . Remark 1 We can also define the predecessor and the successor of the vertex in the order " " ' induced by the cycle. Then, the vertex is of type (or has sign ) if and ) ) ' are both arcs of the digraph, and is of type (or has sign ) if both and are arcs of the digraph.. . Definition 1.6 A Hamilton cycle (resp. circuit) of a digraph is a cycle (resp. circuit) which contains every vertex exactly once. Definition 1.7 We will say that a digraph is decomposable into Hamilton cycles (resp. circuits) if its arcs can be partitioned into Hamilton cycles (resp. circuits)..

(20) will be said to be -crossing if the cycle crosses all

(21) , . ' - and - - . Figure (3) shows examples of -crossing Hamilton cycles in. Definition 1.8 A Hamilton cycle of. the vertices of level and furthermore. 1.2 Results Various results have been obtained on the existence of Hamilton cycles in classical networks (see for example the surveys [2, 7]). For example, it is well-known that any Cayley graph on an abelian group is hamiltonian. Furthermore, it has been conjectured by Alspach [1] that: Conjecture 1 (Alspach) Every connected Cayley graph on an abelian group has a Hamilton decomposition.. . This conjecture has been verified for all connected 4-regular graphs on abelian groups in [6]. It includes in particular the toroidal meshes (grids). For the hypercube, it is also known that is decomposable into Hamilton cycles (see [2, 3]).. . The wrapped Butterfly digraph is actually a Cayley graph (on a non-abelian group) and a line digraph. So, the decomposition into Hamilton cycles (resp. circuits) of this digraph has received some attention. It is well-known that has one Hamilton circuit (see [8, page 465] for a proof in the case or [12]). In [4], Barth and Raspaud proved that the underlying multigraph associated to contains two arc-disjoint Hamilton cycles answering a conjecture of J. Rowley and D. Sotteau [10]. In our terminology, their result can be stated as:. . RR n˚2920.

(22) .

(23) 6. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. Theorem 1.1 (Barth, Raspaud).

(24) is decomposable into Hamilton cycles.. They conjectured that this result can be generalized for any degree: Conjecture 2 (Barth, Raspaud) For. , is decomposable into Hamilton cycles.. . In this paper, we prove the conjecture (2). To do so, we use some techniques introduced in [5] where we studied the decomposition of into Hamilton circuits. In fact, we prove that . is decomposable into -crossing Hamilton cycles. Indeed, the -crossing property, , enables us to prove that the number of combined with the recursive structure of . -crossing arc-disjoint Hamilton cycles that contains can only increase when increases.. Then, we prove mainly that contains arc-disjoint -crossing Hamilton cycles, by cons ) tructing two arc-disjoint -crossing Hamilton cycles using only arcs of slopes 0 and 1 and arcdisjoint Hamilton circuits using arcs of other slopes. The results are summarized in the following theorem:.

(25) . . , - , is decomposable into ) Hamilton circuits and for cycles, *

(26) , is decomposable into Hamilton circuits, for . -

(27) is decomposable into ' Hamilton circuit and Hamilton cycles.. . Theorem 1.2 For. Hamilton. 2 The general construction We give below some additional definitions and properties enabling us to establish lemma (2.2) which is indeed a strengthened version of the inductive lemma of [5]. This lemma is then applied in section (3) to construct inductively the decomposition.. or equivalently the associated perfect .

(28) Definition 2.1 Let be a set of slopes (that is a subset of ). Then, a perfect matching of. " . A family of perfect matchings uses the slopes in if, for any of uses the slopes in if, for any perfect matching of the family, uses the slopes in . be families of perfect matchings. ' Definition 2.2 For , let are said to be compatible if, for each in , the perfect matchings are The families # $ , for %& (' ). arc-disjoint (i.e. !" . 2.1 Families of perfect matchings. . . We will denote a permutation of mapping to by which contains all the arcs . matching of. INRIA.

(29) Hamilton cycle decomposition of the Butterfly network. . 7. . . . . is of type if: Definition 2.4 A family of perfect matchings " ; , , for , , " . for

(30) # is cyclic-potent if and Lemma 2.1 A family of perfect matchings of type ) only if is prime with .. . . Definition 2.3 A family of perfect matchings satisfies the cyclic-potent ) ' ' property if, for any order of composition of the and any set of sign such that , the permutation is cyclic. ,. .

(31)

(32)

(33) :

(34)

(35) a set of families of perfect matchings of type

(36) We will represent by the array:

(37)

(38) &

(39) . . Proof. As the permutations of the family commute, the permutation of definition (2.3). " can be simply expressed as #" . So, this permutation will be cyclic#if and only if is prime " ) ) , , we have . As . with . Here ) So, is clearly prime with if and only if is prime with .. In section (3) we will need some very simple cyclic-potent families of perfect matchings that we give as examples. Families 1 There exist. . compatible cyclic-potent families of perfect matchings: ,. . . '. '. -. . . . . ) )('. '). . . ). ' , )%'. . , These families are cyclic-potent as, applying lemma (2.1), ) )(' ' , which is prime with . These families use all the slopes.. . . )%'). ) . ' :. Families 2 There exist compatible families which use the slopes , ,. . ' ' ,. According to lemma (2.1) they are two compatible cyclic-potent families and they use the slopes ' , .. RR n˚2920.

(40) 8. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. *) compatible cyclic-potent families of perfect matchings exist

(41) , )there ' . One possible solution is given below: % - , the following families can be used: When is odd and - ) - ) ) ' ) ' . Families 3 When using the slopes . When. . is even, we use the following families: -. . . ) )+-. -. . . . ) . . . ). '. ) ). '. and ) ) - ) ) ' , - ) ) ) ' ) ) ' and ) - ) for )even, ) )(' ) ' , which are prime with . ) ' . In both cases, the slopes used are in. These families are cyclic-potent as, applying lemma (2.1), we get:. . for odd, as 2 and 4 are prime with ;.

(42) . 2.2 Inductive construction. ). ' ). . . ). -. . . Lemma 2.2 If admits arc-disjoint cycles and if there exist -crossing Hamilton " ' cyclic-potent families of perfect matchings in , then admits arc-disjoint. -crossing Hamilton cycles.. . . . Proof. Let be an -crossing Hamilton cycle of . As all the levels are equiva lent, we can suppose without loss of generality and for simplicity in the notations , . Let that be a cyclic-potent family of perfect matchings of . The vertices of " ' ! & . Now, we associate can be labeled and with " ' to and a partial digraph in as follows (for an example of such a construction see figure (3)):. .

(43) . . . . ' . . ) ' for each , if the arc .

(44)

(45) . ' " ' belongs to , we put in ' the

(46) ' and ' " ' , which means that to the arc the indices are taken modulo

(47) ' , of where ) ' '

(48) ' is associated the arc in ; . " ' we put the arcs joining to . , . between levels and , of " ' is incident to two arcs of ' . Hence, we With such a definition, each vertex of for , " arc. )(' .. . . '. can define for each vertex a predecessor and a successor on order in a cycle.. . . ' that enables us to prove that we can . . . INRIA.

(49) Hamilton cycle decomposition of the Butterfly network. . ' '

(50) ' ' ) be the predecessor (resp. successor) of in ,

(51) in ' will be ' ' (resp. ' ' ' ' ). . " ) For is a , -crossing Hamilton cycle, vertices or , and , as , are either of type on . . , is of type " , its predecessor (resp. successor) in the cycle is ' ) ' (resp. When ' ' ' ). Then, ' the predecessor (resp. successor) of will be '

(52) ) ' (resp. in . , ); the , will be . (resp. predecessor (resp. successor) of ' ' ' ). . , is of type ) , its predecessor (resp. successor) in is ' ' (resp. ' ' ) ' ). When ' the predecessor (resp. successor) of , will be ' ' (resp. . ); Then, in

(53) will be . , (resp. ' ' )(' ) in ' . the predecessor (resp. successor) of . " ) and , are vertices of type " (resp. ) ) Therefore, when , is of type (resp. ), . ' ' in . Hence, all the vertices of levels , and are crossed by ; furthermore, the sum of the signs of. ' the vertices of of levels , or will be times the sum of the signs of the vertices of of level , , ' is , -crossing (and also -crossing). that is, by hypothesis, , . Hence, '. . 9. ). ' ' . '. For , let (resp. then the predecessor (resp. successor) of. . . . . . . . . . . . . . . . . '. '. Now, we have to prove that is effectively a Hamilton cycle. For this it suffices to prove that. if we start at some vertex , and follow , we meet successively all the vertices of level , and , . Indeed, suppose that before coming back to was on the portion of cycle between . .

(54) . $ , and , . Then, will be on the portion of between and , where $ $ . " ) . ) ) if ) if , (resp. , is of type (resp. ), and , (resp. , is of type " (resp. ). These cases are described on figure (2).. . . . . . '. . . , , be the sequence of vertices of at level , in the or , we will meet successively , , from ' . Following , on. Starting " such a path, we can meet either of type by going from. ) level to level , , in which case we will apply the perfect matching to some , or of type. . where by going from level , to , in which case we will apply to . So onlytheat , product is taken in an order depending on . As all the differ, we can meet again

(55) some , but being cyclic-potent, the values are all distinct. , only after having encountered the So, we meet again vertices of level , . Now, note that we can perform this construction with arc-disjoint , -crossing cycles and compatible cyclic-potent families. From construction, the , -crossing cycles that we will obtain will be Now, let , der we meet them on . . . . . . . . . arc-disjoint.. . . Remark 2 When the , -crossing Hamilton cycles used in the lemma above of , are circuits " " ' all the vertices are of type , and the construction leads to circuits of , giving another proof of the inductive lemma of [5].. RR n˚2920.

(56) 10. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. figure a’. figure a. +. +. x1. +. x2. ax2. 0. +. x2. -. 0 figure c.

(57) . 0. +. -. figure b’. ax1. -. -. -. 0. ax1. x1. ax2. +. n 0 figure c’. . x1. x2. +. n. x1. ax1. figure b. n. ax1. -. x2. ax2. 0 figure d. ax2. 0. -. +. n 0 figure d’. ' ' . '. Figure 2: This figure shows of the four possible cases when we perform the inductive construction. *" ' from . In figure and (resp. and ) the vertices and are of " ). " type (resp. ). Figure and (resp. and ) displays the case where the vertex is of type ). ) " (resp. ) and the vertex is of type (resp. ).. 3. '. .

(58) Decomposition of. into two partial digraphs. is the sum of two partial digraphs Definition 3.1 The Butterfly digraph

(59) defined as follows: and contains the arcs which slopes belong to , ' , contains the arcs which slopes belong to

(60) )(' . We will use a decomposition of. 3.1 Decomposition of. !#"%$'&(*). The proof is inductive on . We start the induction for. . '. ..

(61) , ' is decomposable into Hamilton circuits. ' ,+ , ' is obtained from + by removing the loops Proof. As ' + without the loops contains ) ' and the arcs of slope . Following Tillson [11], we know that % . arc-disjoint Hamilton circuits when . So, using Tillson decomposition, we can label the verLemma 3.1 When. INRIA.

(62) Hamilton cycle decomposition of the Butterfly network. +. . . . 11 '. tices of such that one of the circuits uses all the arcs of slope . By removing it we get ' arc-disjoint Hamilton circuits in .. *) . - , is decomposable into Hamilton circuits. &

(63) , the proposition is proved for ' by lemma (3.1). Then, as - , the As ) Proof. compatible )+' and satisfy cyclic-potent families (3) in section (2.1) use the slopes the (2.2). Hence, we can apply that lemma ) hypothesis arc-disjointof lemma inductively,

(64) . in order to construct Hamilton circuits (see remark (2)) in Proposition 3.1 For. "%$'&(*). 3.2 Decomposition of. is decomposable into -crossing Hamilton cycles. will be denoted by the couples . with Proof. For this proof, the vertices of % ' . We will show that we can build two arc-disjoint -crossing Hamilton , and . . cycles in by using two sets of arcs of defined by the next two rules: Lemma 3.2. . 1. Arcs of. :. . 2. Arcs of. )('

(65) , ' . ,

(66) ' , '

(67) " ' , . if. if :. . . if. if. , . ' . " ' ,

(68) ' )('

(69) , . ' , . . It is easy to verify that and are arc-disjoint. With the arcs (1) of each a dipath as follows:. . . . . , we can define for. " ' ' . " ' , *" ,

(70) ' . *" )

(71) ' . . " )

(72) , . . " )('

(73) ' . *" )('

(74) , . ' are not The dipaths , , are clearly vertex-disjoint. Only the vertices noted in these dipaths. The arcs (2) of. ' as follows: allows us to join the end vertices of the dipaths through the missing vertices. . . RR n˚2920.

(75) 12. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes.

(76) " ) ' " ) ' '

(77) " ) " ) '

(78) " ' " ' ' . ' . . . . . . One can easily check that we have defined a Hamilton cycle. The their extremal vertices in a cyclic way, using only arcs (2) of .. . . dipaths are joined through. . '. . By construction, all the vertices at level are crossed. In order to compute the sign of the vertices ' . . " ' ' at level , we can choose to walk along the cycle in the direction . Therefore, ,. '. " " '. ' all the vertices with are of type and have as sign, while the vertices are ) ) ' ) ) of type and have as sign. So, the sum of the signs is . ,. . . '. To prove that the second set of rules builds a second -crossing Hamilton cycle, it suffices to. notice that we can rewrite this rule up to a permutation of the letters and as being:. Arcs of. (with permutation of . . if if. and ):. . " '

(79) , , . . . ' ' . . ,

(80) ) ' , . Construction 2 is then clearly similar to construction 1; to be convinced, just exchange ' ) ' and replace by in the proof for construction (1). '. . and ,. Hence, and are two arc-disjoint -crossing Hamilton cycles. As the levels are equivalent, the result holds also for level 0. . . . .- into two ' -crossing Hamilton cycles. , is decomposable into -crossing Hamilton cycles. Proposition 3.2 For by the lemma (3.2). Then, we use lemma (2.2) with Proof. The proposition is proved for ' to construct the two compatible cyclic-potent families (2) in section (2.1) which use the slopes , .

(81) . inductively two arcs-disjoint -crossing Hamilton cycles in Figure (3) gives a decomposition of. '. the recursive construction of two -crossing Hamilton cycles in Figure - (3) - gives ' - arc-disjoint . from two -crossing arc-disjoint cycles in. INRIA.

(82) Hamilton cycle decomposition of the Butterfly network. 13. figure b. figure d. figure a. figure c. . . '. .- . We obtained by. Figure 3: Figures and show the two -crossing arc-disjoint Hamilton cycles of .- ' display on figures and , two -crossing arc-disjoint Hamilton cycles in applying lemma (2.2) with the families (2).. 3.3 Global Decomposition We are now ready to prove the main result:. , - , is decomposable into ) Hamilton circuits and for cycles, *

(83) , is decomposable into Hamilton circuits, for . -

(84) is decomposable into ' Hamilton circuit and Hamilton cycles.. Theorem 3.3 For. RR n˚2920. Hamilton.

(85) 14. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes.

(86) . - )

(87)

(88) . Proof. According to propositions (3.1) and (3.2) we have, when , arc-disjoint and arc-disjoint cycles in . So, the result holds in circuits in ( and , an exhaustive computer search shows that is these cases. For decomposable into Hamilton circuits,- and so, for , and are decompo' sable into Hamilton circuits. For , we can construct two -crossing arc-disjoint Hamilton cycles and one arc-disjoint Hamilton circuit in (see figure (4)). Then, we can apply lemma (2.2) with families (1) and the result holds for with .. Figure 4: The decomposition of arc-disjoint Hamilton circuit.. . .- into two ' -crossing arc-disjoint Hamilton cycles and one. ,

(89) is decomposable into Hamilton cycles. Remark 3 We could also have derived theorem (3.4) by proving that, if. + " ' is also decomposable intois decompo sable into -crossing Hamilton cycles, then -crossing The preceding result implies the conjecture of Barth and Raspaud:. Theorem 3.4 For any. and. Hamilton cycles. This can be done by applying lemma (2.2) with the families (1) in section (2.1). But to start the induction we needed to split the Butterfly digraph into two partial digraphs in order . to prove that is decomposable into -crossing Hamilton cycles for and .. . 4 Conclusion. .

(90) . . In this paper we have proved that is always decomposable into Hamilton cycles. However, the problem of decomposing into Hamilton circuits remains open and is considered in [5]. The difficulty in that case is to start the induction. In fact, we conjecture that is decomposable into Hamilton circuits for . Unfortunately such a decomposition is not yet known, even if in [5] we were able to reduce the problem to the case prime and to solve it in many cases. Consequently, we propose as open problem the following conjecture:. . . INRIA.

(91) Hamilton cycle decomposition of the Butterfly network. Conjecture 3 ([5]) For any prime number cuits.. -. ,. 15. is decomposable into Hamilton cir-. Proving this conjecture would completely close the problem of the Hamilton decomposition of the Butterfly network.. Acknowledgements We thank the referees for their helpful remarks.. References [1] B. Alspach. Research problem 59. Discrete Mathematics, 50:115, 1984. [2] B. Alspach, J-C. Bermond, and D. Sotteau. Decomposition into cycles I: Hamilton decompositions. In G. Hahn et al., editor, Cycles and Rays, Proceeding Colloquium Montreál, 1987, NATO ASI Ser. C, pages 9–18, Dordrecht, 1990. Kluwer Academic Publishers. [3] J. Aubert and B. Schneider. Dećomposition de la somme carte´sienne d’un cycle et de l’union de deux cycles en cycles hamiltoniens. Discrete Mathematics, 38:7–16, 1982. [4] D. Barth and A. Raspaud. Two edge-disjoint hamiltonian cycles in the Butterfly graph. Information Processing Letters, 51:175–179, 1994. [5] J-C. Bermond, E. Darrot, O. Delmas, and S. Perennes. Hamilton circuits in directed Butterfly networks. Technical Report RR 95-47, I3S - CNRS URA 1376, 1995. Submitted to Discrete Applied Mathematics. [6] J-C. Bermond, O. Favaron, and M. Maheo. Hamiltonian decomposition of Cayley graphs of degree 4. Journal of Combinatorial Theory, Series B, 46(2):142–153, 1989. [7] S.J. Curran and J.A. Gallian. Hamilton cycles and paths in Cayley graphs and digraphs - a survey. To appear in Discrete Mathematics. [8] F. Thomson Leighton. Introduction to Parallel Algorithms and Architectures: Arrays . Trees . Hypercubes. Computer Science, Mathematics Electrical Engineering. Morgan Kaufmann Publishers, 1992. [9] Jean de Rumeur. Communication dans les re´seaux de processeurs. Collection Etudes et Recherches en Informatique. Masson, 1994. (English version to appear).. + . Journal of Combinatorial Theory,. [10] D. Sotteau and J. Rowley. Private communication. [11] T. Tillson. A Hamiltonian decomposition of Series B, 29:68–74, 1980.. RR n˚2920.

(92) 16. J-C. Bermond, E. Darrot, O. Delmas & S. Perennes. [12] S.A. Wong. Hamilton Cycles and Paths in Butterfly Graphs. Networks, 26(3):145–150, October 1995.. Table of Contents 1 Introduction and notations 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 3 5. 2 The general construction 2.1 Families of perfect matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Inductive construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 6 8. 3 Decomposition of . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Decomposition of . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Decomposition of 3.3 Global Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 10 11 13. 4 Conclusion. 14.

(93) . INRIA.

(94) Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex 1 Unite´ de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unite´ de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. E´diteur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.

(95)