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Octahedral designs
Author(s) -
Linek Václav,
Stevens Brett
Publication year - 2010
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20262
Subject(s) - octahedron , mathematics , combinatorics , order (exchange) , annals , crystallography , crystal structure , chemistry , classics , finance , economics , history
An oriented octahedral design of order v , or OCT( v ), is a decomposition of all oriented triples on v points into oriented octahedra. Hanani [H. Hanani, Decomposition of hypergraphs into octahedra, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, 319, New York Academy of Science, New York, 1979, pp. 260–264.] settled the existence of these designs in the unoriented case. We show that an OCT( v ) exists if and only if v ≡1, 2, 6 (mod 8) (the admissible numbers), and moreover the constructed OCT( v ) are unsplit, i.e. their octahedra cannot be paired into mirror images. We show that an OCT( v ) with a subdesign OCT( U ) exists if and only if v and u are admissible and v ≥ u +4. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:319–327, 2010