Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.     

Extended (2, 4)-designs

In this paper, the concept of an extended (2, 4)-design is introduced. An extended (2, 4)-design is a pair (X, $\text{B}$) where X is a finite set and $\text{B}$ is a collection of 4-tuples of not necessarily distinct elements of X, such that every pair of not necessarily distinct elements of X is contained in exactly one member of $\text{B}$. It is shown that an extended (2,4)-design of order n exists for every positive integer n except n = 6, 8 and 9. Several inequivalent designs of order n are obtained.     

Switching for 2-designs

Designs, Codes and Cryptography - In this paper, we introduce a switching for 2-designs, which defines a type of trade. We illustrate this method by applying it to some symmetric...     

Schematic 4-designs

We study 4-designs with three intersection numbers. By the Cameron-Delsarte theorem, the blocks form a symmetric three-class association scheme. This imposes strong restrictions on the parameters of such designs. We calculate the eigenvalues of the association scheme from the design parameters and determine all admissible parameters with at most 1000 points. An infinite family of admissible parameters is discovered. Designs with small admissible parameters exist and are related to the quadratic residue codes.     

Splitter Theorems for 4-Regular Graphs

Let ${\Phi_{k,g}}$ be the class of all k-edge connected 4-regular graphs with girth of at least g. For several choices of k and g, we determine a set ${\mathcal{O}_{k,g}}$ of graph operations, for which, if G and H are graphs in ${\Phi_{k,g}}$ , G ≠ H, and G contains H as an immersion, then some operation in ${\mathcal{O}_{k,g}}$ can be applied to G to result in a smaller graph G′ in ${\Phi_{k,g}}$ such that, on one hand, G′ is immersed in G, and on the other hand, G′ contains H as an immersion.     

On 2-designs

Denote by M_v the set of integers b for which there exists a 2-design (linear space) with v points and b lines. M_v is determined as accurately as possible. On one hand, it is shown for v > v₀ that M_v contains the interval [v + p + 1, v + p + q ? 1]. On the other hand for v of the form p² + p + 1 it is shown that the interval [v + 1, v + p ? 1] is disjoint from M_v; and if v > v₀ and p is of the form q² + q, then an additional interval [v + p + 1, v + p + q ? 1] is disjoint from M_v.     

色数等于选择数的4-正则图

设Hn(n≥5)表示一个图:以1,2,．．．,n为顶点,两个点i和j是相邻的当且仅当|i-j|≤2,其中加法取模n．这篇文章证明了,Hn的色数等于它的选择数．结果被用于刻画最大度至多2的图的列表全色数．     

Miscellaneous classification results for 2-designs

An orderly algorithm combined with clique searching is used to show that there are—up to isomorphism, in all cases—325,062 resolvable 2-(16,4,2) designs with 339,592 resolutions, 19,072,802 2-(13,6,5) designs, and 15,111,019 2-(14,7,6) designs. Properties of the classified designs are further discussed.     

Halving Steiner 2-designs

A Steiner 2-design S(2,k,v) is said to be halvable if the block set can be partitioned into two isomorphic sets. This is equivalent to an edge-disjoint decomposition of a self-complementary graph G on v vertices into K_ks. The obvious necessary condition of those orders v for which there exists a halvable S(2,k,v) is that v admits the existence of an S(2,k,v) with an even number of blocks. In this paper, we give an asymptotic solution for various block sizes. We prove that for any k?5 or any Mersenne prime k, there is a constant number v₀ such that if v>v₀ and v satisfies the above necessary condition, then there exists a halvable S(2,k,v). We also show that a halvable S(2,2ⁿ,v) exists for over a half of possible orders. Some recursive constructions generating infinitely many new halvable Steiner 2-designs are also presented.     

Non-symmetric 2-designs modulo 2

Necessary conditions are obtained for the existence of a 2 – (v, k, ) design, for which the block intersection sizess ₁,s ₂, ...,s _n satisfys ₁ s ₂ ... s _n s (mod 2 ^e ), wheree is odd. These conditions are obtained by combining restrictions on the Smith Normal Form of the incidence matrix of the design with some well known properties of self-orthogonal binary codes with all weights divisible by 4.Research done at AT&T Bell Laboratories.     

A new family of 4-designs

We construct a family of simple 4-designs with parameters 4 – (2 ^f + 1, 6, 10),f odd.     

Infinite families of 2-designs and 3-designs from linear codes

The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the largest $t$ for which an infinite family of $t$-designs is derived directly from a linear or nonlinear code is $t=3$. Sporadic $4$-designs and $5$-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of $2$-designs and $3$-designs from linear codes. The parameters of some known $t$-designs are also derived. In addition, many conjectured infinite families of $2$-designs are also presented.     

4-Regular Vertex-Transitive Tilings of E 3

{There exist precisely 149 topological types of semipolytopal tile-transitive tilings of E ³ by ``extetrahedra' (obtained from tetrahedra by introducing certain new vertices of degree 2 ). Dualization gives rise to 149 types of 4-regular vertex-transitive tilings. The 4-coordinated networks carried by these tilings are closely related to crystal structures such as zeolites or diamond. These results are obtained using ``combinatorial tiling theory.'} Received February 12, 1999, and in revised form September 21, 1999. Online publication May 15, 2000.