A packing problem its application to Bose's families

We propose and study the following problem: given X ⊂ Z_n, construct a maximum packing of dev X (the development of X), i.e., a maximum set of pairwise disjoint translates of X. Such a packing is optimal when its size reaches the upper bound . In particular, it is perfect when its size is exactly equal to i.e. when it is a partition of Z_n. We apply the above problem for constructing Bose's families. A (q, k) Bose's family (BF) is a nonempty family F of subsets of the field GF(q) such that: (i) each member of F is a coset of the kth roots of unity for k odd (the union of a coset of the (k - 1)th roots of unity and zero for k even); (ii) the development of F, i.e., the incidence structure , is a semilinear space. A (q, k)-BF is optimal when its size reaches the upper bound . In particular, it is perfect when its size is exactly equal to ; in this case the (q, k)-BF is a (q, k, 1) difference family and its development is a linear space. If the set of (q, k)-BF's is not empty, there is a bijection preserving maximality, optimality, and perfectness between this set with the set of packings of dev X, where X is a suitable -subset of Z_n, for k odd, for k even. © 1996 John Wiley & Sons, Inc.     

A non-existence result on cyclic cycle-decompositions of the cocktail party graph

We prove that in every cyclic cycle-decomposition of K_2n−I (the cocktail party graph of order 2n) the number of cycle-orbits of odd length must have the same parity of n(n−1)/2. This gives, as corollaries, some useful non-existence results one of which allows to determine when the two table Oberwolfach Problem OP(3,2?) admits a 1-rotational solution.     

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Designs, Codes and Cryptography - Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with...     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an i‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph K_v to be i‐perfect if for every edge [x, y] of K_v there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ‐decomposition (Γ‐factorization) of K_v that is i‐perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge‐colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i‐perfect Γ‐decomposition of K_v provided that Γ is an i‐equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009     

Perfect Cayley Designs as Generalizations of Perfect Mendelsohn Designs

We introduce the concept of a Perfect Cayley Design(PCD) that generalizes that of a Perfect Mendelsohn Design (PMD) as follows. Given anadditive group H, a (v, H, 1)-PCDis a pair whereX is a v-set and isa set of injective maps fromH toX with the property that for any pair (x,y)of distinct elements of X and any h H - {0} there is exactly one B such that B(h')=x, B(h')=yandh'-h'=h for suitable h',h' H.It is clear that a (v,Z_k,1)-PCD simply is a(v, k, 1)-PMD.This generalization has concretemotivations in at least one case. In fact we observe thattriplewhist tournaments may be viewed as resolved(v,Z ₂ ² ,1)-PCD's but not, in general, as resolved(v, 4, 1)-PMD's.We give four composition constructionsfor regular and 1-rotational resolved PCD's. Two of them make use of differencematrices and contain, asspecial cases, previous constructions for PMD's by Kageyama andMiao [15] and for Z-cyclic whist tournaments by Anderson,Finizio and Leonard [5]. The other two constructions succeed wheresometimes difference matrices fail and their applications allow us to get new PMD's, new Z-cyclic directed whist tournaments and newZ-cyclic triplewhist tournaments.The whist tournaments obtainable with the last twoconstructions are decomposable into smaller whist tournaments.We show this kind of tournaments useful in practice whenever, at theend of a tournament, some confrontations between ex-aequo players areneeded.     

A powerful method for constructing difference families and optimal optical orthogonal codes

In this paper we proceed in the way indicated by R. M. Wilson for obtaining simple difference families from finite fields [28]. We present a theorem which includes as corollaries all the known direct techniques based on Galois fields, and provides a very effective method for constructing a lot of new difference families and also new optimal optical orthogonal codes.By means of our construction—just to give an idea of its power—it has been established that the only primesp<10⁵ for which the existence of a cyclicS(2, 9,p) design is undecided are 433 and 1009. Moreover we have considerably improved the lower bound on the minimumv for which anS(2, 15,v) design exists.     

Some Remarks on the Rule of the Middle Third

The so-called rule of the middle third states that an inclined force applied at the top of a vertical pillar with rectangular cross-section must intersect the bottom within the middle third of the height of the rectangle in order that the normal stress on the base is one-signed. This rule has been extended by Michell (1900) to the case of a plane elastic wedge loaded at its vertex. We here study plane elastic pillars with other profiles, like a trapezoid, the plane region bounded by two branches of an equilateral hyperbola, a blunt pillar. The result is that the rule is only partially valid.     

Small quasimultiple affine and projective planes: Some improved bounds

We improve the known bounds on r(n): = min {λ| an (n², n, λ)-RBIBD exists} in the case where n + 1 is a prime power. In such a case r(n) is proved to be at most n + 1. If, in addition, n − 1 is the product of twin prime powers, then r(n) ${\ \le \ }{n \over 2}$. We also improve the known bounds on p(n): = min{λ| an (n² + n + 1, n + 1, λ)-BIBD exists} in the case where n² + n + 1 is a prime power. In such a case p(n) is bounded at worst by but better bounds could be obtained exploiting the multiplicative structure of GF(n² + n + 1). Finally, we present an unpublished construction by M. Greig giving a quasidouble affine plane of order n for every positive integer n such that n − 1 and n + 1 are prime powers. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 337–345, 1998