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.     

Equipartite gregarious 6- and 8-cycle systems

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 gregarious 3-cycle systems are 3-GDDs, and necessary and sufficient conditions for their existence are known (see for instance the CRC Handbook of Combinatorial Designs, 1996, C.J. Colbourn, J.H. Dinitz (Eds.), Section III 1.3). The cases of equipartite and of almost equipartite 4-cycle systems were recently dealt with by Billington and Hoffman. Here, for both 6-cycles and for 8-cycles, we give necessary and sufficient conditions for existence of a gregarious cycle decomposition of the complete equipartite graph K_n(a) (with n parts, n?6 or n?8, of size a).     

Varieties of quasigroups arising from 2-perfect m-cycle systems

For m = 6 and for all odd composite integers m, as well as for all even integers m 10 that satisfy certain conditions, 2-perfect m-cycle systems are constructed whose quasigroups have a homomorphism onto quasigroups which do not correspond to a 2-perfect m-cycle systems. Thus it is shown that for these values of m the class of quasigroups arising from all 2-perfect m-cycle systems does not form a variety.     

Twofold 2-perfect bowtie systems with an extra property

A bowtie is a closed trail whose graph consists of two 3-cycles with exactly one vertex in common. A 2-fold bowtie system of order n is an edge-disjoint decomposition of 2K _n into bowties. A 2-fold bowtie system is said to be 2-perfect provided that every pair of distinct vertices is joined by two paths of length 2. It is said to be extra provided these two paths always have distinct midpoints. The extra property guarantees that the two paths x, a, y and x, b, y between every pair of vertices form a 4-cycle (x, a, y, b), and that the collection of all such 4-cycles is a four-fold 4-cycle system. We show that the spectrum for extra 2-perfect 2-fold bowtie systems is precisely the set of all n ?? 0 or 1 (mod 3), ${n\,\geqslant\,6}$ . Additionally, with an obvious definition, we show that the spectrum for extra 2-perfect 2-fold maximum packings of 2K _n with bowties is precisely the set of all n ?? 2 (mod 3), ${n\,\geqslant\,8}$ .     

Perfect 7-cycle systems

Perfect m-cycle systems are defined. In this paper, it is proven that the class of perfect 7-cycle systems is a variety of quasigroups, and a defining set of identities for this variety is given.     

The complete spectrum of 6-cycle systems of l(kn)

In this article, it is shown that a decomposition of the line graph of the complete graph on n vertices into cycles of length 6 exist if and only if n ≢ 3 (mod 4). © 1995 John Wiley & Sons, Inc.     

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 . We then construct a 5-cycle system of order 10n+1 which can be embedded in a pentagon triple system of order 30n+1 and also construct a 5-cycle system of order 10n+5 which can be embedded in a pentagon triple system of order 30n+15, with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.     

Almost topological dynamical systems

For the notion of finitary isomorphism, which arises in many examples in ergodic theory, we prove some basic theorems about invariants, representations and the central limit theorem in shift spaces.     

On tight 6-cycle decompositions of complete 3-uniform hypergraphs

The complete 3-uniform hypergraph of order v has a vertex set V of size v and the set of all 3-element subsets of V as its edge set. A tight 6-cycle is a hypergraph with vertex set $\{a,b,c,d,e,f\}$ and edge set $\{\{a,b,c\},\{b,c,d\},\{c,d,e\},\{d,e,f\},\{e,f,a\},\{f,a,b\}\}$. We show that there exists a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of a tight 6-cycle if and only if $v\equiv 1$, 2, 10, 20, 28, or $29(\mathrm{mod}36)$.     

ν-perfect groups

Let F_∞ be a free group on a countable set {x₁, x₂, …} and ν be a variety of groups, defined by the set of outer commutators V, in the free generators x_i's.The paper is devoted to give the complete structure of a ν-covering of ν-perfect groups. Fur thermore necessary and sufficient conditions for the universality of a ν-central extension by a group and its ν-covering group will be presented.     

Almost periodic transformable difference systems

We analyse almost periodic homogeneous linear difference systems whose coefficient matrices belong to the so-called transformable groups. We introduce the notion of weakly transformable groups and we use this notion to obtain generalizations of known results about non-almost periodic solutions of considered systems.     

Almost all triangle-free triple systems are tripartite

A triangle in a triple system is a collection of three edges isomorphic to {123,124,345}. A triple system is triangle-free if it contains no three edges forming a triangle. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is triangle-free. We prove that almost all triangle-free triple systems with vertex set [n] are tripartite. Our proof uses the hypergraph regularity lemma of Frankl and R?dl [13], and a stability theorem for triangle-free triple systems due to Keevash and the second author [15].