Embedding 4-cycle systems into octagon triple systems
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.
