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This paper studies techniques of finding hamiltonian paths and cycles in hypercubes and dense sets of hypercubes. This problem is, in general, easily solvable but here the problem was modified by the requirement that a set of edges has to be used in such path or cycle. The main result of this paper says that for a given n, any sufficiently large ------------------

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A set A of vertices of a hypercube is called balanced if |{A ∈ A | |A| ≡ 0 mod 2}| = |{A ∈ A | |A| ≡ 1 mod 2}|. We prove that for every natural number n there exists a natural number 1 (n) such that for every hypercube Q with dim(Q) 1, 2, . . . , n} is a balanced set.

If a regular caterpillar is a spanning subgraph of a hypercube , then it has 2 n Ϫ 1 legs for some n and its length is 2 m for some m . We prove the converse statement : for every n there exists m 0 such that for every m у m 0 a regular caterpillar of length 2 m with 2 n Ϫ 1 legs is a spanning subgraph of a hypercube of dimension n ϩ m .÷ 1997 Academic Press Limited . I NTRODUCTIONParallelism is one of the most important topics in computer science' even though many theoretical parallel models of computation are not technically realizable . This inspires investigation of simulation techniques between dif ferent parallel models . A hypercube is a typical technical reasonable parallel model , and therefore simulations of dif ferent parallel models by a hypercube are studied in many papers . Since any simulation , in a broad sense , can be described by some type of embedding between graphs , this motivates us to study such embeddings-see the excellent survey papers due to Sudborough and Monien [13 , 14] .It is well known that many types of graphs modeling a parallel architecture have an embedding into hypercubes ; for example , rings , two-dimensional meshes , higherdimensional meshes , hexagons and almost complete binary trees-see Havel and Mora ´ vek [11] , Nebesky ´ [15] and Wagner [16] . The first papers investigating this problem were inspired by switching circuits and coding theory , and they established some properties of subgraphs of hypercubes-see Djokovic ä [2] , Firsov [5] or Garey and Graham [6] . Afrati et al . [1] showed that it is NP-complete to decide whether a given graph is a subgraph of a hypercube . This result was generalized by Wagner and Corneil [17] , who proved that a decision as to whether a given tree is a subgraph of a given hypercube is NP-complete .We restrict ourselves to an embedding of special trees-caterpillars-into hypercubes . A characterization of several types of caterpillars , which are spanning subgraphs of hypercubes , was given by T . Dvor ä a ´ k , F . Harary