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Equitable resolvable coverings
Abstract
In an earlier article, Willem H. Haemers has determined the minimum number of parallel classes in a resolvable 2-(qk,k,1) covering for all k ≥ 2 and q = 2 or 3. Here, we complete the case q = 4, by construction of the desired coverings using the method of simulated annealing. Secondly, we look at equitable resolvable 2-(qk,k,1) coverings. These are resolvable coverings which have the additional property that every pair of points is covered at most twice. We show that these coverings satisfy k < 2q − , and we give several examples. In one of these examples, k > q. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 113–123, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10024
Given five positive integers $v, m,k,\lambda$ and $t$ where $v \geq k \geq t$
and $v \geq m \geq t,$ a $t$-$(v,k,m,\lambda)$ general covering design is a
pair $(X,\mathcal{B})$ where $X$ is a set of $v$ elements (called points) and
$\mathcal{B}$ a multiset of $k$-subsets of $X$ (called blocks) such that every
$m$-subset of $X$ intersects (is covered by) at least $\lambda$ members of
$\mathcal{B}$ in at least $t$ points. In this article we present new
constructions for general covering designs and we generalize some others. By
means of these constructions we will be able to obtain some new upper bounds on
the minimum size of such designs.
For prime power q, we determine the minimum number of parallel classes in a resolvable 2-(kq, k, 1) covering for almost all values of k. In September 1995 Jan van Haastrecht gave a dinner on the occasion of his retirement. He had invited twenty colleagues. The host and the guests were to be seated at three tables each of which had seven places. The diner consisted of five courses. Jan wanted the guests to change places between the courses in such a way that everyone would meet everyone else (Jan included) at least once at some table. At that time no solution was known and so a few couples didn't meet.
A minimum covering of pairs by triples in a 6n-element set contains 6n 2 triples. Can such a covering be resolvable? A. Assaf, E. Mendelsohn and D. R. Stinson [Util. Math. 32, 67-74 (1987; Zbl 0649.05024)] showed that this is not possible for n<3, and that for n≥3 such a resolvable covering exists if n∉{6,7,8,10,1,13,14,17,22}. In the present paper, we show that such resolvable coverings exist for these nine values of n.
Let V be a finite set of v elements. A covering of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at least one member of F. For fixed v and k, the covering problem is to determine the number of blocks in any minimum covering. A minimum covering is resolvable if we can partition the blocks into classes (called resolution classes) such that every element is contained in precisely one block of each class. A resolvable minimum covering of the pairs of V by k-subsets is denoted by RC(v, k). In this article, we show that there exist RC(v, 4) for v ≡ 0 (mod 4) except for v = 12 and possibly for v ∈ {104, 108, 116, 132, 156, 164, 204, 212, 228, 276}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 431–450, 1998
A t-cover of the finite projective space PG(d,q) is a setS of t-dimensional subspaces such that any point of PG(d,q) is contained in at least one element ofS. In Theorem 1 a lower bound for the cardinality of a t-coverS in PG(d,q) is obtained and in Theorem 2 it is shown that this bound is best possible for all positive integers t,d and for any prime-power q.
A t-cover of a finite projective space ℙ is a set of t-dimensional subspaces covering all points of ℙ. Beutelspacher [1] constructed examples of t-covers and proved that his examples are of minimal cardinality. We shall show that all examples of minimal cardinality “look
like” the examples of Beutelspacher.
A t Gamma (v; m; k; ) covering design is a pair (X; A), where X is a set of v elements (called points) and A is a multiset of k-subsets of X (called blocks), such that every m-subset of X intersects at least members of A in at least t points. It is required that v k t and v m t. Such a covering design gives an upper bound on C (v; m; k; t), the number of blocks in a minimal t Gamma (v; m; k; ) covering design. In this paper it is shown how simulated annealing, a probabilistic method for solving combinatorial optimization problems, can be used to construct covering designs. Implementation of this method is discussed. The method is used to calculate new upper bounds on C 1 (v; t; k; t) for k 9, v 13. A new exact value, C 1 (11; 4; 6; 4) = 32, is obtained. 1 Introduction Let v k t and v m t. A t Gamma (v; m; k; ) covering design is a pair (X; A), where X is a set of v elements (called points) and A is a multiset of k-subsets of X (called blocks), such that every m-s...
The CRC Handbook of Combinatorial Designs
- D. R. Stinson
- Lamken
- Haemers