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, Selda
Yazici, Emine Sule
Lindner, Curt
Title Embedding 4-cycle systems into octagon triple systems
Journal name Utilitas Mathematica   Check publisher's open access policy
ISSN 0315-3681
Publication date 2009-07-01
Year available 2009
Sub-type Article (original research)
Open Access Status Not yet assessed
Volume 79
Start page 99
End page 106
Total pages 8
Place of publication University of Durban, South Africa
Publisher University of KwaZulu-Natal, Durban, South Africa
Language eng
Subject C1
970101 Expanding Knowledge in the Mathematical Sciences
010104 Combinatorics and Discrete Mathematics (excl. Physical Combinatorics)
Abstract 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.
Formatted abstract
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.
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
2010 Higher Education Research Data Collection
 
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Citation counts: TR Web of Science Citation Count  Cited 5 times in Thomson Reuters Web of Science Article | Citations
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Created: Thu, 03 Sep 2009, 17:55:07 EST by Mr Andrew Martlew on behalf of Faculty of Science