{'fraction_non_alphanumeric': 0.04931972789115646, 'fraction_numerical': 0.027210884353741496, 'mean_word_length': 4.3545454545454545, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On Vertex Partitions of Hypercubes by Isometric Trees

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.