单纯有向三元系的嵌入

本文给出并证明了单纯有向三元系嵌入的充要条件，从而完整地解决了这一问题．     

Smallest defining sets of directed triple systems

A directed triple system of order v, , is a pair (V,B) where V is a set of v elements and B is a collection of ordered triples of distinct elements of V with the property that every ordered pair of distinct elements of V occurs in exactly one triple as a subsequence. A set of triples in a D is a defining set for D if it occurs in no other on the same set of points. A defining set for D is a smallest defining set for D if D has no defining set of smaller cardinality. In this paper we are interested in the quantity

     

On the existence of nearly Kirkman triple systems with subsystems

It is proved in this paper that the necessary and sufficient conditions for the existence of an incomplete nearly Kirkman triple system INKTS(u, v) are u ≡ v ≡ 0 (mod 6), u ≥ 3v. As a consequence, we obtain a complete solution to the embedding problem for nearly Kirkman triple systems.      

Self-converse large sets of pure directed triple systems

An $LPDTS(v,\lambda )$ is a collection of $\frac{3(v?2)}{\lambda }$ pairwise disjoint $PDTS(v,\lambda )s$ on the same set of v elements. An $LPDT{S}^{?}(v)$ is a special $LPDTS(v,1)$ which contains exactly $\frac{v?2}{2}$ converse hexads of $PDTS(v)s$. In this paper, we mainly discuss the existence of an $LPDT{S}^{?}(v)$ and get the following conclusions: (1) there exists an $LPDT{S}^{?}(v)$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 4$ except possibly $v=30$. (2) There exists an $LPDTS(v,\lambda )$ with index $\lambda \equiv 2,4\mathrm{mod 6}$ if and only if $v\equiv 0,4\mathrm{mod 6},v\ge 2\lambda +2,v\equiv 2\mathrm{mod}\mathit{\lambda }$ except possibly $v=30$.     

Embeddings of pure mendelsohn triple systems and pure directed triple systems

It is proved in this article that the necessary and sufficient conditions for the embedding of a λ-fold pure Mendelsohn triple system of order v in λ-fold pure Mendelsohn triple of order u are λu(u ? 1) ≡ 0 (mod 3) and u ? 2v + 1. Similar results for the embeddings of pure directed triple systems are also obtained. © 1995 John Wiley & Sons, Inc.     

Extended directed triple systems

Let {n;b₂,b₁} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b₂ and the number of blocks of the form [b,a,a] or [a,a,b] is b₁. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b₂,b₁} is b₁≠1, 0?b₂+b₁?n and

(1)

for ;

(2)

for .

     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Large sets of extended directed triple systems with odd orders

For three types of triples, unordered, cyclic and transitive, the corresponding extended triple, extended triple system and their large set are introduced. The spectrum of LEDTS(v) for even v has been given in our paper (Liu and Kang (2009) [9]). In this paper, we shall discuss the existence problem of LEDTS(v) for odd v and give the almost complete conclusion: there exists an LEDTS(v) for any positive integer v≠4 except possible v=95,143,167,203,215.     

A conjecture on small embeddings of partial Steiner triple systems

A well‐known, and unresolved, conjecture states that every partial Steiner triple system of order u can be embedded in a Steiner triple system of order υ for all υ ≡ 1 or 3, (mod 6), υ ≥ 2u + 1. However, some partial Steiner triple systems of order u can be embedded in Steiner triple systems of order υ <2u + 1. A more general conjecture that considers these small embeddings is presented and verified for some cases. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 313–321, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10017     

On list coloring Steiner triple systems

We study the list chromatic number of Steiner triple systems. We show that for every integer s there exists n₀=n₀(s) such that every Steiner triple system on n points STS(n) with n≥n₀ has list chromatic number greater than s. We also show that the list chromatic number of a STS(n) is always within a log n factor of its chromatic number. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 314–322, 2009     

The spectrum for large sets of pure directed triple systems

An LPDTS(ν) is a collection of 3(ν-2) disjoint pure directed triple systems on the same set ofνelements. It is showed in Tian's doctoral thesis that there exists an LPDTS(ν) forν=0,4 (mod 6),ν≥4. In this paper, we establish the existence of an LPDTS(ν) forν= 1,3 (mod 6),ν> 3. Thus the spectrum for LPDTS(ν) is completely determined to be the set {ν:ν= 0, 1 (mod 3),ν≥4}.     

The spectrum for overlarge sets of directed triple systems

A directed triple system of order v,denoted by DTS(v,λ),is a pair(X,B)where X is a v- set and B is a collection of transitive triples on X such that every ordered pair of X belongs toλtriples of B.An overlarge set of disjoint DTS(v,λ),denoted by OLDTS(v,λ),is a collection{(Y\{y},A_i)}_i, such that Y is a(v 1)-set,each(Y\{y},A_i)is a DTS(v,λ)and all A_i's form a partition of all transitive triples of Y.In this paper,we shall discuss the existence problem of OLDTS(v,λ)and give the following conclusion:there exists an OLDTS(v,λ)if and only if eitherλ=1 and v≡0,1(mod 3),orλ=3 and v≠2.     

Embeddings of resolvable mendelsohn triple systems

In this article it is shown that any resolvable Mendelsohn triple system of order u can be embedded in a resolvable Mendelsohn triple system of order v iff v≥ 3u, except possibly for 71 values of (u,v). © 1993 John Wiley & Sons, Inc.     

Nilpotent block designs I: Basic concepts for Steiner triple and quadruple systems

This paper discusses the concepts of nilpotence and the center for Steiner Triple and Quadruple Systems. The discussion is couched in the language of block designs rather than algebras. Nilpotence is closely connected to the well known doubling and tripling constructions for these designs. A sample result: a point p in an STS is projective if every triangle containing p generates the 7-element Fano plane; the p-center of the STS is the set of all projective points and is a projective geometry over GF(2). © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 157–171, 1999