1,721,098 research outputs found
Cycle decompositions with a sharply vertex transitive automorphism group
No full textIn some recent papers the method of partial differences introduced by the author [4] was very helpful in the construction of cyclic cycle systems. Here we use this method for the purpose of constructing, more generally, cycle decompositions with a sharply vertex transitive automorphism group not necessarily cyclic
Pairwise balanced designs from finite fields
No full textDeveloping previous work by R.C. Bose, R.M. Wilson, M. Greig and the author, we give a general construction for difference families over fnite fields. All the DFs constructed before were generated by a single initial base block so that they were uniform. Here we also consider DFs generated by two or more initial base blocks. These DFs give rise to pairwise balanced designs which of course are balanced block designs exactly when the initial base blocks have constant size
Pairwise balanced designs from finite fields
No full textAbstractDeveloping previous work by R.C. Bose, R.M. Wilson, M. Greig and the author, we give a general construction for difference families over finite fields. All the DFs constructed before were generated by a single initial base block so that they were uniform. Here we also consider DFs generated by two or more initial base blocks. These DFs give rise to pairwise balanced designs which of course are balanced block designs exactly when the initial base blocks have constant size
Bruck-Ryser Abstract Theorem and Symmetric Designs
No full textIn [3] U. Ott introduced a Bruck-Ryser abstract theorem concerning "lattices of an R-module". As one of the applications of this theorem, he obtained a new proof of the Bruck-Ryser theorem for finite projective planes and studid "p-curves" in a projective plane. In this paper, we apply the Bruck-Ryser abstract theorem in order to give a new proof of the Bruck-Ryser-Chowla theorem for symmetric (v.k,lambda)-designs with (v,k,lambda)=1, and to study "p-curves" in arbitrary symmetric designs
Some regular Steiner 2-designs with block size 4
No full textWe give a constructive and very simple proof of a theorem by P. L. Check and C. J. Colbourn [Discrete Math. 133 (1994), no. 1-3, 285--289] stating the existence of a cyclic -BIBD (i.e. regular over ) for any prime . We extend the theorem to primes , although in this case the construction is not explicit. Anyway, for all these primes , we explicitly construct a regular -BIBD over
On resolvable difference families
No full textA Steiner 2-design is said to be G-invariantly resolvable if admits an automorphism group G and a resolution invariant under G. Introducing and studying resolvable difference families, we characterize the class of G-invariantly resolvable Steiner 2-designs arising from relative difference families over G. Such designs have been already studied by Genma, Jimbo, and Mishima [13] in the case in which G is cyclic. Developing their results, we prove that any (p, k, 1)-DF (p prime) whose base blocks exactly cover (p–1)/k(k–1) distinct cosets of the k-th roots of unity (mod p), leads to a -invariantly resolvable cyclic (kp,k,1)-BBD. This induced us to propose several constructions for DF's having this property. In such a way we prove, in particular, the existence of a -invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1000
Edge-colourings characterizing a class of Cayley Graphs and a New Characterization of Hypercubes
No full textCayley graphs on the group , in particular the hypercube , are considered. A characterization of them as graphs having suitable edge-colouring properties is given
On disjoint (v,k,k-1) difference families
No full textA disjoint (v, k, k − 1) difference family in an additive group G is a partition of G{0}
into sets of size k whose lists of differences cover, altogether, every non-zero element of G exactly k − 1 times. The main purpose of this paper is to get the literature on this topic in order, since some authors seem to be unaware of each other’s work. We show, for instance, that a couple of heavy constructions recently presented as new, had been given in several equivalent forms over the last forty years.We also show that they can be quickly derived from a general nearring theory result which probably passed unnoticed by design theorists and that we restate and reprove, more simply, in terms of differences. This result can be exploited to get many infinite classes of disjoint (v, k, k −1) difference families; here, as an example, we present an infinite class coming from the Fibonacci sequence. Finally, we will prove that if all prime factors of v are congruent to 1 modulo k, then there exists a disjoint (v, k, k − 1) difference family in every group, even non-abelian, of order v
On a property of symmetric designs of order n = 2 (mod 4)
No full textUsing the Bruck-Ryser-Chowla theorem and the identity it is proved that, for any symmetric design of order ,
Clique-colourings characterizing Hamming graphs
No full textHamming graph are characterized by means of suitable clique-colourings
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