Realizations for direct constructions of resolvable Steiner 2‐designs with block size 5

We consider direct constructions due to R. J. R. Abel and M. Greig, and to M. Buratti, for ({ν},5,1) balanced incomplete block designs. These designs are defined using the prime fields F_p for certain primes p, are 1‐rotational over G ⊕ F_p where G is a group of order 4, and are also resolvable under certain conditions. We introduce specifications to the constructions and, by means of character sum arguments, show that the constructions yield resolvable designs whenever p is sufficiently large. © 2000 John Wiley & Sons, Inc. J Combin Designs 8:207–217, 2000     

On E(s2)‐optimal and minimax‐optimal supersaturated designs with 20 rows and 76 columns

We derive a previously unknown lower bound of 41 for the frequency of of an E(s²)‐optimal and minimax‐optimal supersaturated design (SSD) with 20 rows and 76 columns. This is accomplished by an exhaustive computer search that uses the combinatorial properties of resolvable 2 − (20, 10, 36) designs and the parallel class intersection pattern method. We also classify all nonisomorphic E(s²)‐optimal 4‐circulant SSDs with 20 rows and .     

A complete classification of (12,4,3)‐RBIBDs

Using a parallel class by parallel class backtracking algorithm we classify (12,4,3)‐RBIBDs. The size of the search space is reduced by combinatorial arguments. Five nonisomorphic RBIBDs are found and listed. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 385–400, 2001     

一种构造多水平超饱和设计的新方法

超饱和设计在搜索试验中有重要应用.自上世纪九十年代迄今, 超饱和设计的研究取得了丰硕成果,研究的设计从二水平发展到多水平, 进而到混合水平;构造方法从巧妙的构思到利用组合理论进行系统构造,进而到各种算法的开发; 对于超饱和设计的评优准则也有更深入的认识.本文介绍一种构造多水平超饱和设计的新方法. 这种方法简单易行,很容易构造出具有优良性质的多水平超饱和设计, 有很强的实用性.     

The Existence of Partitioned Generalized Balanced Tournament Designs with Block Size 3

A generalized balanced tournament design, GBTD(n, k), defined on a kn-set V, is an arrangement of the blocks of a (kn, k, k – 1)-BIBD defined on V into an n × (kn – 1) array such that (1) every element of V is contained in precisely one cell of each column, and (2) every element of V is contained in at most k cells of each row. Suppose we can partition the columns of a GBTD(n, k) into k + 1 sets B1, B2,..., Bk + 1 where |Bi| = n for i = 1, 2,..., k – 2, |Bi| = n–1 for i = k – 1, k and |Bk+1| = 1 such that (1) every element of V occurs precisely once in each row and column of Bi for i = 1, 2,..., k – 2, and (2) every element of V occurs precisely once in each row and column of Bi Bk+1 for i = k – 1 and i = k. Then the GBTD(n, k) is called partitioned and we denote the design by PGBTD(n, k). The spectrum of GBTD(n, 3) has been completely determined. In this paper, we determine the spectrum of PGBTD(n,3) with, at present, a fairly small number of exceptions for n. This result is then used to establish the existence of a class of Kirkman squares in diagonal form.     

A construction of resolvable nested 3‐designs

A method of constructing resolvable nested 3‐designs from an affine resolvable 3‐design is proposed with one example. © 2004 Wiley Periodicals, Inc.     

Decomposable super‐simple NRBIBDs with block size 4 and index 6

Necessary conditions for the existence of a super‐simple, decomposable, near‐resolvable ‐balanced incomplete block design (BIBD) whose 2‐component subdesigns are both near‐resolvable ‐BIBDs are (mod ) and . In this paper, we show that these necessary conditions are sufficient. Using these designs, we also establish that the necessary conditions for the existence of a super‐simple near‐resolvable ‐RBIBD, namely (mod ) and , are sufficient. A few new pairwise balanced designs are also given.     

A complete solution to the existence problem for 1‐rotational k‐cycle systems of Kv

The necessary and sufficient conditions for the existence of a 1‐rotational k‐cycle system of the complete graph K_v are established. The proof provides an algorithm able to determine, directly and explicitly, an odd k‐cycle system of K_v whenever such a system exists. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 283–293, 2009     

The spectrum of α‐resolvable designs with block size four

Abstact: An α‐resolvable BIBD is a BIBD with the property that the blocks can be partitioned into disjoint classes such that every class contains each point of the design exactly α times. In this paper, we show that the necessary conditions for the existence of α‐resolvable designs with block size four are sufficient, with the exception of (α, ν, λ) = (2, 10, 2). © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 1–16, 2001     

Enumeration of resolvable 2‐(10, 5, 16) and 3‐(10, 5, 6) designs

A backtracking over parallel classes with a partial isomorph rejection (PIR) is carried out to enumerate the resolvable 2‐(10,5,16) designs. Computational results show that the inclusion of PIR reduce substantially the CPU time for the enumeration of all designs. We prove first some results, which enable us to restrict the search space. Since every resolvable 2‐(10,5,16) design is also a resolvable 3‐(10,5,6) design and vice versa, the latter designs are also enumerated. There are 27, 121, 734 such designs with automorphism groups whose order range from 1 to 1,440. From these, designs 2,006,690 are simple. © 2004 Wiley Periodicals, Inc.     

1‐rotational k‐factorizations of the complete graph and new solutions to the Oberwolfach problem

We consider k‐factorizations of the complete graph that are 1‐rotational under an assigned group G, namely that admit G as an automorphism group acting sharply transitively on all but one vertex. After proving that the k‐factors of such a factorization are pairwise isomorphic, we focus our attention to the special case of k = 2, a case in which we prove that the involutions of G necessarily form a unique conjugacy class. We completely characterize, in particular, the 2‐factorizations that are 1‐rotational under a dihedral group. Finally, we get infinite new classes of previously unknown solutions to the Oberwolfach problem via some direct and recursive constructions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 87–100, 2008     

Degree‐ and Orbit‐Balanced Γ‐Designs When Γ Has Five Vertices

A Γ‐design of the complete graph is a set of subgraphs isomorphic to Γ (blocks) whose edge‐sets partition the edge‐set of . is balanced if the number of blocks containing x is the same number of blocks containing y for any two vertices x and y. is orbit‐balanced, or strongly balanced, if the number of blocks containing x as a vertex of a vertex‐orbit A of Γ is the same number of blocks containing y as a vertex of A, for any two vertices x and y and for every vertex‐orbit A of Γ. We say that is degree‐balanced if the number of blocks containing x as a vertex of degree d in Γ is the same number of blocks containing y as a vertex of degree d in Γ, for any two vertices x and y and for every degree d in Γ. An orbit‐balanced Γ‐design is also degree‐balanced; a degree‐balanced Γ‐design is also balanced. The converse is not always true. We study the spectrum for orbit‐balanced, degree‐balanced, and balanced Γ‐designs of when Γ is a graph with five vertices, none of which is isolated. We also study the existence of balanced (respectively, degree‐balanced) Γ‐designs of which are not degree‐balanced (respectively, not orbit‐balanced).     

New Infinite Families of 2‐Edge‐Balanced Graphs

A graph G of order n is called t‐edge‐balanced if G satisfies the property that there exists a positive λ for which every graph of order n and size t is contained in exactly λ distinct subgraphs of isomorphic to G. We call λ the index of G. In this article, we obtain new infinite families of 2‐edge‐balanced graphs.     

Rotational k‐cycle systems of order v < 3k; another proof of the existence of odd cycle systems

We give an explicit solution to the existence problem for 1‐rotational k‐cycle systems of order v < 3k with k odd and v ≠ 2k + 1. We also exhibit a 2‐rotational k‐cycle system of order 2k + 1 for any odd k. Thus, for k odd and any admissible v < 3k there exists a 2‐rotational k‐cycle system of order v. This may also be viewed as an alternative proof that the obvious necessary conditions for the existence of odd cycle systems are also sufficient. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 433–441, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10061     

1‐rotationally resolvable 4‐cycle systems of 2Kv

In this article, it is shown that there exists a 1‐rotationally resolvable 4‐cycle system of 2K_υ if and only if υ ≡ 0 (mod 4). To prove that, some special sequences of integers are utilized. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 116–125, 2002; DOI 10.1002/jcd.10006     

There is no 2‐(22, 8, 4) block design

In this article, we show that a 2‐(22,8,4) design does not exist. This result was obtained by a computer search. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 262–267, 2007     

Exceptions and characterization results for type‐1 λ‐designs

Let $X$ be a finite set with $v$ elements, called points and $\beta$ be a family of subsets of $X$, called blocks. A pair $\left(X,\beta \right)$ is called $\lambda$‐design whenever $\mid \beta \mid =\mid X\mid$ and

• 1. for all $B_i,B_j\in \beta ,i\ne j,\mid B_i\cap B_j\mid =\lambda$;
• 2. for all $B_j\in \beta ,\mid B_j\mid =k_j>\lambda$, and not all $k_j$ are equal.

The only known examples of $\lambda$‐designs are so‐called type‐1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$‐designs are type‐1. Let $r,r^{*}?(r>r^{*})$ be replication numbers of a $\lambda$‐design $D=\left(X,\beta \right)$ and $g=\text{gcd}\left(r?1,r^{*}?1\right),m=\text{gcd}\left(\left(r?r^{*}\right)∕g,\lambda \right)$, and $m\prime =m$, if $m$ is odd and $m\prime =m∕2$, otherwise. For distinct points $x$ and $y$ of $D$, let $\lambda \left(x,y\right)$ denote the number of blocks of $X$ containing $x$ and $y$. We strengthen a lemma of S.S. Shrikhande and N.M. Singhi and use it to prove that if $\frac{r\left(r?1\right)}{\left(v?1\right)}?\frac{k\left(r?r^{*}\right)}{m\prime \left(v?1\right)}$ are not integers for $k=1,2,\text{…},m\prime ?1$, then $D$ is type‐1. As an application of these results, we show that for fixed positive integer $\theta$ there are finitely many nontype‐1 $\lambda$‐designs with $r=r^{*}+\theta$. If $r?r^{*}=27$ or $r?r^{*}=4p$ and $r^{*}\ne {\left(p?1\right)}^2$, or $v=7p+1$ such that $p?1,13\left(\mathrm{mod}21\right)$ and $p?4,9,19,24\left(\mathrm{mod}35\right)$, where $p$ is a positive prime, then $D$ is type‐1. We further obtain several inequalities involving $\lambda (x,y)$, where equality holds if and only if D is type‐1.     

On the existence of (3,λ)‐semiframes of type 3u

A (k,λ)‐semiframe of type g^u is a (k,λ)‐group‐divisible design of type g^u (??, ??, ??), in which the collection of blocks ?? can be written as a disjoint union ??=??∪?? where ?? is partitioned into parallel classes of ?? and ?? is partitioned into holey parallel classes, each holey parallel class being a partition of ??\G_j for some G_j∈??. In this paper, we shall prove that the necessary conditions for (3,λ)‐semiframes of type 3^u are also sufficient with one exception. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 253–265, 2009     

Some new results on 1‐rotational 2‐factorizations of the complete graph

It is known that a necessary condition for the existence of a 1‐rotational 2‐factorization of the complete graph K_2n+1 under the action of a group G of order 2n is that the involutions of G are pairwise conjugate. Is this condition also sufficient? The complete answer is still unknown. Adapting the composition technique shown in Buratti and Rinaldi, J Combin Des, 16 (2008), 87–100, we give a positive answer for new classes of groups; for example, the groups G whose involutions lie in the same conjugacy class and having a normal subgroup whose order is the greatest odd divisor of |G|. In particular, every group of order 4t+2 gives a positive answer. Finally, we show that such a composition technique provides 2‐factorizations with a rich group of automorphisms. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 237–247, 2010