Halving Steiner 2-designs
Halving Steiner 2-designs
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DOI：
10.1016/j.disc.2006.09.005
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发表时间：
2007-06
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影响因子：
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通讯作者：
Yuichiro Fujiwara
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文献类型：
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作者：
Yuichiro Fujiwara
如果块集可以划分为两个同构集，则称Steiner 2-设计S（2,k,v）是可分的。这等价于v个顶点上的自互补图G的边不相交分解成Kks。对于存在可分S（2,k,v）的阶v，其明显的必要条件是，v允许存在具有偶数块的S（2,k,v）。本文给出了不同块大小的渐近解。证明了对于任意k≤5或任意Mersenne素数k，存在一个常数v0，使得v>v0和v满足上述必要条件，则存在一个可分的S（2,k,v）。我们还证明了一个可分的S（2,2n,v）存在于超过一半的可能阶。并给出了一些产生无穷多个新的可减半斯坦纳2型设计的递归结构。
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 Kks. 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 v0such that if v>v0and 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.