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When n=2m-1 Ramras [J. Combin. Theory Ser. B, 54 (1992), pp. 239–248] proved, by a counting argument, that for any isometrically embedded tree T on n edges in Qn there exists a group of translations G such that {g(T);g∈G} is a vertex partition of Qn. Considering a more general context we are able to give an explicit construction of G and can construct nongroup vertex partitions by isometric trees. We also extend this problem to vertex partition of Qn′ by translates of an isometrically embedded tree on n=2m-1 edges for any n′≥n.

On Vertex Partitions of Hypercubes by Isometric Trees