5-cycle Systems Research Articles

5-cycle systems of [formula omitted] and [formula omitted

In Bryant et al. (1996) it is shown that there exists a 5-cycle system of Kv+u−Kv if and only if the obvious necessary conditions are satisfied. One purpose of this paper is to extend this result by providing the necessary and sufficient conditions for the existence of a 5-cycle system of λKv+u−λKv.It was shown in Asplund et al. (2013) that if there exists a 5-cycle system of λKv, then there exists a 5-cycle system of (λ+m)Kv+1−λKv if and only if v is (λ,5)-admissible. The second purpose of this paper is to extend this result by removing the requirement that there exists a 5-cycle system of λKv. In doing so, we will show that there exists a 5-cycle system of (λ+m)Kv+1−λKv if and only if the obvious necessary conditions are satisfied, except possibly in two cases.

Enclosings of [formula omitted]-fold 5-cycle systems for [formula omitted

A k-cycle system of a multigraph G is an ordered pair (V,C) where V is the vertex set of G and C is a set of k-cycles, the edges of which partition the edges of G. A k-cycle system of λKv is known as a λ-fold k-cycle system of order v. A k-cycle system of λKv(V,C) is said to be enclosed in a k-cycle system of (λ+m)Kv+u(V∪U,P) if C⊂P and u,m≥1. In this paper the enclosing problem for 5-cycle systems is settled in the general situation where the three parameters λ, m, and v are allowed to vary freely and u is constrained to the difficult case of adding two vertices. New graph theoretic approaches are introduced to handle this situation developing an avenue of research that is of interest in its own right.

Embedding 5-cycle systems into pentagon triple systems

We show that the spectrum for pentagon triple systems is the set of all n ≡ 1 , 15 , 21 or 25 ( mod 30 ) . We then construct a 5-cycle system of order 10 n + 1 which can be embedded in a pentagon triple system of order 30 n + 1 and also construct a 5-cycle system of order 10 n + 5 which can be embedded in a pentagon triple system of order 30 n + 15 , with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.

Equipartite Gregarious 5-Cycle Systems and Other Results

A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite 3-cycle systems are 3-GDDs (and so are automatically gregarious), and necessary and sufficient conditions for their existence are known. The cases of equipartite gregarious 4-, 6- and 8-cycle systems have also been dealt with (using techniques that could be applied in the case of any even length cycle). Here we give necessary and sufficient conditions for the existence of a gregarious 5-cycle decomposition of the complete equipartite graph Km(n) (in effect the first odd length cycle case for which the gregarious constraint has real meaning). In doing so, we also define some general cyclic constructions for the decomposition of certain complete equipartite graphs into gregarious p-cycles (where p is an odd prime).

Minimum embedding of balanced P4-designs into 5-cycle systems

A 5-cycle system on v+ w points embeds a balanced P 4-design on v points if there is a subset of v points on which the 5-cycles induce the blocks of a balanced P 4-design. The mininum possible such w is v( mod 30) w v≡4,16 v−1 3 v≡7,10,22,25 v+5 3 v≡1,13,28 v+11 3 v≡19 ( mod 30) v+17 3 We shall show this minimum value of w is attained.

5-Cycle systems with holes

Recently the generalized Doyen-Wilson problem of embedding a 5-cycle system of order u in one of order v was completely solved. However it is often useful to solve the more general problem of the existence of a 5-cycle system of order v with a hole of size u. In this paper we completely solve this problem.

The Doyen—Wilson theorem extended to 5-cycles

In this paper it is proved that any 5-cycle system of order u can be embedded in a 5-cycle system of order ν iff ν > 3u 2 and ν ≡ 1 or 5 (mod 10) .