Strongly 2-perfect cycle systems and their quasigroups

A recent result of Bryant and Lindner shows that the quasigroups arising from 2-perfect m-cycle systems form a variety only when m = 3, 5 and 7. Here we investigate the situation in the case where the distance two cycles are required to be in the original system.

Lambda-fold 3-perfect 9-cycle systems

The spectrum for the decomposition of lambda K-v into 3-perfect 9-cycles is found for all lambda > 1. (The case lambda = 1 was dealt with in an earlier paper by the authors and Lindner.) The necessary conditions for the existence of a suitable decomposition turn out to be sufficient.

Varieties of algebras arising from K-perfect m-cycle systems

A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m is an element of {3, 5} only, and that the algebras arising from {1, 2}-perfect m-cycle systems form a variety for m is an element of {3, 5, 7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety. (c) 2006 Elsevier B.V. All rights reserved.

2-perfect directed m-cycle systems can be equationally defined for m=3,4, and 5 only

It is shown that quasigroups constructed using the standard construction from 2-perfect directed m-cycle systems are precisely the finite members of a variety if and only if m=3, 4 or 5.

On single laws for varieties of quasigroups associated with extended cycle systems

It has been previously shown by Lindner and Rodger that quasigroups associated with 2-perfect extended m-cycle systems can be equationally defined if and only if m is an element of {3, 5, 7}. In this paper we present a single identity for each such m which is equivalent to the identities given for these varieties.

Irreducible 2-fold cycle systems

In this paper, it is shown that for any pair of integers (m, n) with 4 ≤ m ≤ n, if there exists an m-cycle system of order n, then there exists an irreducible 2-fold m-cycle system of order n, except when (m, n) = (5,5). A similar result has already been established for the case of 3-cycles. © 2005 Wiley Periodicals, Inc.

(m,n)-cycle systems

We describe a method which, in certain circumstances, may be used to prove that the well-known necessary conditions for partitioning the edge set of the complete graph on an odd number of vertices (or the complete graph on an even number of vertices with a 1-factor removed) into cycles of lengths m(1),m(2),...,m(t) are sufficient in the case \{m(1), m(2), ..., m(t)}\=2. The method is used to settle the case where the cycle lengths are 4 and 5. (C) 1998 Elsevier Science B.V. All rights reserved.

The mu-way intersection problem for m-cycle systems

An m-cycle system of order upsilon is a partition of the edge-set of a complete graph of order upsilon into m-cycles. The mu -way intersection problem for m-cycle systems involves taking mu systems, based on the same vertex set, and determining the possible number of cycles which can be common to all mu systems. General results for arbitrary m are obtained, and detailed intersection values for (mu, m) = (3, 4), (4, 5),(4, 6), (4, 7), (8, 8), (8, 9). (For the case (mu, m)= (2, m), see Billington (J. Combin. Des. 1 (1993) 435); for the case (Cc,m)=(3,3), see Milici and Quattrochi (Ars Combin. A 24 (1987) 175. (C) 2001 Elsevier Science B.V. All rights reserved.

The metamorphosis of lambda-fold 4-wheel systems into lambda-fold 4-cycle systems

A 4-wheel is a simple graph on 5 vertices with 8 edges, formed by taking a 4-cycle and joining a fifth vertex (the centre of the 4-wheel) to each of the other four vertices. A lambda -fold 4-wheel system of order n is an edge-disjoint decomposition of the complete multigraph lambdaK(n) into 4-wheels. Here, with five isolated possible exceptions when lambda = 2, we give necessary and sufficient conditions for a lambda -fold 4-wheel system of order n to be transformed into a lambda -fold Ccyde system of order n by removing the centre vertex from each 4-wheel, and its four adjacent edges (retaining the 4-cycle wheel rim), and reassembling these edges adjacent to wheel centres into 4-cycles.

Skolem-type difference sets for cycle systems

Cyclic m-cycle systems of order v are constructed for all m greater than or equal to 3, and all v = 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle system of K-v - F for all m greater than or equal to 3, and all v = 2(mod 2m).

Dissection of the role of pili and type 2 and 3 secretion systems in adherence and biofilm formation of an atypical enteropathogenic escherichia coli strain

Atypical enteropathogenic Escherichia coli (aEPEC) strains are diarrheal pathogens that lack bundle-forming pilus production but possess the virulence-associated locus of enterocyte effacement. aEPEC strain 1551-2 produces localized adherence (LA) on HeLa cells; however, its isogenic intimin (eae) mutant produces a diffuse-adherence (DA) pattern. In this study, we aimed to identify the DA-associated adhesin of the 1551-2 eae mutant. Electron microscopy of 1551-2 identified rigid rod-like pili composed of an 18-kDa protein, which was identified as the major pilin subunit of type 1 pilus (T1P) by mass spectrometry analysis. Deletion of fimA in 1551-2 affected biofilm formation but had no effect on adherence properties. Analysis of secreted proteins in supernatants of this strain identified a 150-kDa protein corresponding to SslE, a type 2 secreted protein that was recently reported to be involved in biofilm formation of rabbit and human EPEC strains. However, neither adherence nor biofilm formation was affected in a 1551-2 sslE mutant. We then investigated the role of the EspA filament associated with the type 3 secretion system (T3SS) in DA by generating a double eae espA mutant. This strain was no longer adherent, strongly suggesting that the T3SS translocon is the DA adhesin. In agreement with these results, specific anti-EspA antibodies blocked adherence of the 1551-2 eae mutant. Our data support a role for intimin in LA, for the T3SS translocon in DA, and for T1P in biofilm formation, all of which may act in concert to facilitate host intestinal colonization by aEPEC strains. ©2013, American Society for Microbiology.

5-cycle systems of K-v-F with a hole

Recently the problem of the existence of a 5-cycle system of K-v with a hole of size u was completely solved. In this paper we prove necessary and sufficient conditions on v and u for the existence of a 5-cycle system of K-v - F, with a hole of size u.

Embeddings of m-cycle systems and incomplete m-cycle systems: m<=14

In this paper we completely settle the embedding problem for m-cycle systems with m less than or equal to 14. We also solve the more general problem of finding m-cycle systems of K-v - K-u when m is an element of {4,6,7,8,10,12,14}.