# Embedding 4-cycle systems into octagon triple systems

Billington, Elizabeth J., Kucukcifci, Selda, Yazici, Emine Sule and Lindner, Curt (2009) Embedding 4-cycle systems into octagon triple systems. Utilitas Mathematica, 79 99-106.

Author Billington, Elizabeth J.Kucukcifci, SeldaYazici, Emine SuleLindner, Curt Embedding 4-cycle systems into octagon triple systems Utilitas Mathematica   Check publisher's open access policy 0315-3681 2009-07-01 2009 Article (original research) Not yet assessed 79 99 106 8 University of Durban, South Africa University of KwaZulu-Natal, Durban, South Africa eng C1970101 Expanding Knowledge in the Mathematical Sciences010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics) An octagon triple is the graph consisting of the four triangles (triples) {a, b, c}, {c, d, e}, {e, f, g}, and {g, h, a}, where a, b, c, d, e, f, g and h are distinct. The 4-cycle (a, c, e,g) is called an inside 4-cycle. An octagon triple system of order n is a pair (X, O), where O is a collection of edge disjoint octagon triples which partitions the edge set of K with vertex set X. Let (X, O) be an octagon triple system and let P be the collection of inside 4-cycles. Then (X, P) is a partial 4-cycle system of order n. It is not possible for (X, P) to be a 4-cycle system (not enough 4-cycles). So the problem of determining for each n the smallest octagon triple system whose inside 4-cycles contain a 4-cycle system of order 8n + 1 is immediate. The object of this note is to determine the spectrum for octagon triple systems and to construct for every n a 4-cycle system of order k = 8n + 1 that can be embedded in the inside 4-cycles of some octagon triple system of order approximately 3k. This is probably not the best possible embedding (the best embedding is approximately 2k + 1), but it is a good start. An octagon triple is the graph consisting of the four triangles (triples) {a, b, c}, {c, d, e}, {e, f, g}, and {g, h, a}, where a, b, c, d, e, f, g and h are distinct. The 4-cycle (a, c, e,g) is called an inside 4-cycle. An octagon triple system of order n is a pair (X, O), where O is a collection of edge disjoint octagon triples which partitions the edge set of Kn with vertex set X. Let (X, O) be an octagon triple system and let P be the collection of inside 4-cycles. Then (X, P) is a partial 4-cycle system of order n. It is not possible for (X, P) to be a 4-cycle system (not enough 4-cycles). So the problem of determining for each n the smallest octagon triple system whose inside 4-cycles contain a 4-cycle system of order 8n + 1 is immediate. The object of this note is to determine the spectrum for octagon triple systems and to construct for every n a 4-cycle system of order k = 8n + 1 that can be embedded in the inside 4-cycles of some octagon triple system of order approximately 3k. This is probably not the best possible embedding (the best embedding is approximately 2k + 1), but it is a good start. C1 Confirmed Code UQ

 Document type: Journal Article Article (original research) School of Mathematics and Physics 2010 Higher Education Research Data Collection

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