Embedding 4-cycle systems into octagon triple systems - Billington, Elizabeth J.; Küçükçifçi, Selda; Yazici, Emine Šule; Lindner, Curt (Utilitas Mathematica, 2009)
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.