6-cycle Systems Research Articles

Six-cycle systems

AbstractIn this article, we attempt to survey all that has been written about 6-cycle systems. In addition, several new results, including many enumeration results, are included here for the first time. Our survey concludes with a list of open problems.

Block colourings of 6-cycle systems

Let \(\Sigma=(X,\mathcal{B})\) be a \(6\)-cycle system of order \(v\), so \(v\equiv 1,9\mod 12\). A \(c\)-colouring of type \(s\) is a map \(\phi\colon\mathcal {B}\rightarrow \mathcal{C}\), with \(C\) set of colours, such that exactly \(c\) colours are used and for every vertex \(x\) all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(\frac{v-1}{2}=qs+r\), with \(q, r\geq 0\). \(\phi\) is equitable if for every vertex \(x\) the set of the \(\frac{v-1}{2}\) blocks containing \(x\) is partitioned in \(r\) colour classes of cardinality \(q+1\) and \(s-r\) colour classes of cardinality \(q\). In this paper we study bicolourings and tricolourings, for which, respectively, \(s=2\) and \(s=3\), distinguishing the cases \(v=12k+1\) and \(v=12k+9\). In particular, we settle completely the case of \(s=2\), while for \(s=3\) we determine upper and lower bounds for \(c\).

Squashing minimum coverings of 6-cycles into minimum coverings of triples

A 6-cycle is said to be squashed if we identify a pair of opposite vertices and name one of them with the other (thereby turning the 6-cycle into a pair of triples with a common vertex). A 6-cycle can be squashed in six different ways. The spectrum for 6-cycle systems that can be squashed into Steiner triple systems has been determined by Lindner et al.(J Comb Des 22:189–195, 2014). The squashing of maximum packings of Kn with 6-cycles into maximum packings of Kn with triples has also been fully dealt with by Lindner et al. (Squashing maximum packings of 6-cycles into maximum packings of triples (submitted)). The object of this paper is to extend these results to minimum coverings of Kn with 6-cycles into minimum coverings of Kn with triples. We give a complete solution.

Complete sets of metamorphoses: Twofold 4-cycle systems into twofold 6-cycle systems

Let (X,C) denote a twofold k-cycle system with an even number of cycles. If these k-cycles can be paired together so that: (i) each pair contains a common edge; (ii) removal of the repeated common edge from each pair leaves a (2k−2)-cycle; (iii) all the repeated edges, once removed, can be rearranged exactly into a collection of further (2k−2)-cycles; then this is a metamorphosis of a twofold k-cycle system into a twofold (2k−2)-cycle system. The existence of such metamorphoses has been dealt with for the case of 3-cycles (Gionfriddo and Lindner, 2003) [3] and 4-cycles (Yazıcı, 2005) [7].If a twofold k-cycle system (X,C) of order n exists, which has not just one but has k different metamorphoses, from k different pairings of its cycles, into twofold (2k−2)-cycle systems, such that the collection of all removed double edges from all k metamorphoses precisely covers 2Kn, we call this a complete set of twofold paired k-cycle metamorphoses into twofold (2k−2)-cycle systems.In this paper, we show that there exists a twofold 4-cycle system (X,C) of order n with a complete set of metamorphoses into twofold 6-cycle systems if and only if n≡0,1,9,16 (mod 24), n≠9.

Lambda-fold 2-perfect 6-cycle systems in equipartite graphs

A 6-cycle system of a graph G is an edge-disjoint decomposition of G into 6-cycles. Graphs G, for which necessary and sufficient conditions for existence of a 6-cycle system have been found, include complete graphs and complete equipartite graphs. A 6-cycle system of G is said to be 2-perfect if the graph formed by joining all vertices distance 2 apart in the 6-cycles is again an edge-disjoint decomposition of G, this time into 3-cycles, since the distance 2 graph in any 6-cycle is a pair of disjoint 3-cycles.Necessary and sufficient conditions for existence of 2-perfect 6-cycle systems of both complete graphs and complete equipartite graphs are known, and also of λ-fold complete graphs. In this paper, we complete the problem, giving necessary and sufficient conditions for existence of λ-fold 2-perfect 6-cycle systems of complete equipartite graphs.

A solution to the forest leave problem for partial 6-cycle systems

In this paper, we find necessary and sufficient conditions for the existence of a 6-cycle system of K n − E( F) for every forest F of K n .

Partitionable perfect cycle systems with cycle lengths 6 and 8

The spectrum for the decomposition of K r into partitionable 2-perfect 6-cycle systems and partitionable 2-perfect 8-cycle systems is determined, except for four isolated cases. (The nonpartitionable cases were done by Lindner, Phelps and Rodger (1991) and by Adams and Billington (1994), respectively.) Moreover, the spectra for both 3-perfect 8-cycle systems and partitionable 3-perfect 8-cycle systems are also determined.

The spectrum for lambda-fold 2-perfect 6-cycle systems

The spectrum for the decomposition of λK v into 2-perfect 6-cycle systems is found for all λ > 1 and for two outstanding cases when λ = 1, completing work done by Lindner, Phelps and Rodger in the case λ = 1.

Non-isomorphic 2-perfect 6-cycle systems of order 13

Running a computer search for new, cyclic, 2-perfect 6-cycle systems of order 13 and constructing the quasigroups which arise from such systems enabled the author to establish that there are at most two such non-isomorphic systems. Then by using two-variable laws of the quasigroups it is shown that there are exactly two non-isomorphic 2-perfect 6-cycle systems of order 13 which are cyclic.

The spectrum for 2-perfect 6-cycle systems

Recently, the spectrum problem for 2-perfect m-cycle systems has been studied by several authors. In this paper we find the spectrum for 2-perfect 6-cycle systems with two possible exceptions. The connection between these systems and quasigroups satisfying some 2 variable identities is discussed.