Edge Decompositions of Hypercubes by Paths and by Cycles Michel Mollard, Mark Ramras To cite this version: Michel Mollard, Mark Ramras. Edge Decompositions of Hypercubes by Paths and by Cycles. Graphs and Combinatorics, Springer Verlag, 2015. hal-00698983v3 HAL Id: hal-00698983 https://hal.archives-ouvertes.fr/hal-00698983v3 Submitted on 5 Sep 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Edge Decompositions of Hypercubes by Paths and by Cycles Michel Mollard∗ Institut Fourier 100, rue des Maths 38402 St Martin d’H`eres Cedex FRANCE [email protected] and Mark Ramras Department of Mathematics Northeastern University Boston, MA 02115, USA [email protected] September 5, 2013 Abstract If H is isomorphic to a subgraph of G, we say that H divides G if there exist embeddings θ1, θ2, . , θk of H such that {{E(θ1(H)),E(θ2(H)),...,E(θk(H))} is a partition of E(G). For purposes of simplification we will often omit the embeddings, saying that we have an edge decomposition by copies of E(H). Many authors have studied this notion for various subgraphs of hypercubes. We continue such a study in this paper. ∗CNRS Universit´eJoseph Fourier 1 1 Introduction and Preliminary Results Definition 1 If H is isomorphic to a subgraph of G, we say that H divides G if there exist embeddings θ1, θ2, . , θk of H such that {{E(θ1(H)),E(θ2(H)),...,E(θk(H))} is a partition of E(G). Ramras [8] has defined a more restrictive concept. Definition 2 A fundamental set of edges of a graph G is a subset of E(G) whose translates under some subgroup of the automorphism group of G par- tition E(G). Edge decompositions of graphs by subgraphs have a long history. For example, there is a Steiner triple system of order n if and only if the complete graph Kn has an edge-decomposition by K3. In 1847 Kirkman [5] proved that for a Steiner triple system to exist it is necessary that n ≡ 1 (mod 6) or n ≡ 3 (mod 6). In 1850 he proved the converse holds also [6]. Theorem 1 A Steiner system of order n ≥ 3 exists if and only if n ≡ 1 (mod 6)) or n ≡ 3 (mod 6). In more modern times (1964) G. Ringel [11] stated the following conjec- ture, which is still open. Conjecture 1 If T is a fixed tree with m edges then K2m+1 is edge-decomposable into 2m + 1 copies of T . By Qn we mean the n-dimensional hypercube. We regard its vertex set, V (Qn), as P({1, 2, . , n}), the set of subsets of {1, 2, . , n}. Two vertices x and y are considered adjacent (so x, y ∈ E(Qn)) if | x∆y | = 1, where ∆ denotes the symmetric difference of the two subsets x and y.(V (Qn), ∆) is Zn isomorphic as a group to ( 2 , +). Occasionally, when convenient, we shall use the vector notation for vertices; thus x and y are adjacent precisely when they differ in exactly one component. Note that for k < n, P({1, 2, . , k}) ⊂ P({1, 2, . , n}) so that V (Qk) ⊂ V (Qn). In fact, from the definition of adjacency, it follows that Qk is an induced subgraph of Qn. Beginning in the early 1980’s, interest in hypercubes (and similar hypercube- like networks such as “cube-connected cycles” and “butterfly” networks) in- creased dramatically with the construction of massively parallel-processing 2 computers, such as the “Connection Machine” whose architecture is that of the 16-dimensional hypercube, with 216 = 65, 536 processors as the vertices. Problems of routing message packets simultaneously along paths from one processor to another led to an interest in questions of edge decompositions of E(Qn) by paths. An encyclopedic discussion of this and much more can be found in [7]. In [8] we have shown that if G is a subgroup of Aut(Qn) and for all g ∈ G, with g = id (where id denotes the identity element), g(E(H)) ∩ E(H) = ∅, then there is a packing of these translates of E(H) in Qn, i.e. they are pair- n−1 wise disjoint. If, in addition, | E(H) | · | G | = n · 2 = | E(Qn) |, then the translates of E(G) by the elements of G yield an edge decomposition of Qn. In [8] it is shown that every tree on n edges can be embedded in Qn as a fundamental set. (This result for edge decompositions was obtained inde- pendently by Fink [3]). In [9] this is extended to certain trees and certain cycles on 2n edges. Decompositions of Qn by k-stars are proved for all k ≤ n in [2]. Recently, Wagner and Wild [12] have constructed, for each value of n−1 n, a tree on 2 edges that is a fundamental set for Qn. The structure of Aut(Qn) is discussed in [8]. For each subset A of {1, 2, . , n}, the comple- menting automorphism σA is defined by σA(x) = A∆{x}. Another type of automorphism arises from the group of permutations Sn of {1, 2, . , n}. For x = {x1, x2, . , xm} ⊆ {1, 2, . , n} and θ ∈ Sn we denote by ρθ(x) the ver- tex {θ(x1), θ(x2), . , θ(xm)}. The mapping ρθ : V (Qn) −→ V (Qn) defined in this way is easily seen to belong to Aut(Qn). Every automorphism in Aut(Qn) can be expressed uniquely in the form σA ◦ ρθ, where this notation means that we first apply ρθ. Note: ρθ ◦ σA = σθ(A) ◦ ρθ. To avoid ambiguity in what follows we make this definition: Definition 3 By Pk, the “k-path”, we mean the path with k edges. Questions n−1 (1) For which k dividing n · 2 does Pk divide Qn? n−1 (2) For which k dividing n · 2 does Ck, the cycle on k edges, divide Qn? (3) For those k for which the answer to either (1) or (2) is “yes”, is the edge set used in the decomposition a fundamental set for Qn? We begin this introductory section with some examples. In later sections we prove a variety of results relating to these questions, and in the final section we summarize our findings. Example 1 3 Let T be the 2-star (= the 2-path) contained in Q3 with center 000, and leaves 100, 010. Then G = {id, σ123, σ1ρ(123), σ12ρ(132), σ3ρ(132), σ23ρ(123)} is a (cyclic) subgroup of Aut(Q3) of order 6, and the 6 translates of T under G yield an edge decomposition of Q3. 2 Note, however, that G does not work for the 2-star T ′, whose center is 000 and whose leaves are 100 and 001. The subgroup which works for this ′ 2-star is G = {id, σ123, σ1ρ(132), σ13ρ(123), σ2ρ(123), σ23ρ(132)}. Example 2 P6 does not divide Q3. For since Q3 has 12 edges, if P6 did divide Q3 then Q3 would have an edge-decomposition consisting of 2 copies of P6. The degree sequence (in decreasing order) of each P6 is 2, 2, 2, 2, 2, 1, 1, 0, whereas Q3, of course, is 3-regular. Thus the vertex of degree 0 in one P6 would require a degree of 3 in the other, which is impossible. 2 Example 3 P4 does not divide Q3. Since P4 has 4 edges, we would need 3 copies of P4 (1) (2) for an edge-decomposition of Q3. Call the three copies of P4 P ,P , and (3) P . At each vertex v of Q3, P1≤i≤3 degP (i) (v) = 3. Label the vertices of Q3 (1) (v1) to (v8) such that the degree sequence of P , is decreasing. Consider the 3 × 8 array degP (i)(vj) . The first row is thus 2 2 2 1 1 0 0 0. In the second and third rows, in order for the column sums to be 3, there must be exactly 3 1’s (and 3 0’s) in the first 3 columns. Similarly, in the last 3 columns there must be exactly 3 1’s (and 3 0’s). Thus in the second and third rows we have at least 6 1’s, and so at least one of these rows must have at least 3 1’s. But each row is a permutation of the first, which has only 2 1’s. Contradiction. Hence P4 does not divide Q3. 2 Example 4 Since Q3 is 3-regular, the 4-star is not a subgraph. The other tree on 4 edges does divide Q3. Let T be the 3-star centered at 000 union the edge 001, 101. Let G =< σ23ρ(123) >, which is a cyclic subgroup of Aut(Q3) of order 3. A straight-forward calculation shows that the translates of T under G form an edge decomposition of Q3. 2 4 Proposition 1 For k ≥ 3,P2k does not divide Q2k+1. Proof. Suppose that k ≥ 3, and suppose that P2k divides Q2k+1. The matrix 2k+1 (aiv) formed by the degree sequences of copies of P2k has 2 columns, and (2k + 1) · 22k/2k = (2k + 1)2k rows.