四圈系统的稀疏性
Sparseness of 4-cycle systems
摘要：
利用Steiner三元系上的Erdensys r-稀疏猜想中的稀疏性概念，研究了4-圈系统的构形回避问题.一个v阶4-圈系统4CS（v）称为r-稀疏系统，如果对于任意满足1 < j < r+1的整数j，它不包含由j个4-圈组成的构形，而j个4-圈的并集恰好包含j+3个顶点。如果一个r-稀疏4CS（v）也没有两个共享对角线的4-圈上的配置的副本，称为双菱形，我们说它是严格r-稀疏的。本文证明了对每个容许阶v，存在严格4-稀疏的4CS（v）.我们还证明了对任意正整数r > 1和足够大的整数v，存在一个常数c，使得存在v阶严格r-稀疏4-圈填充，且cv^2为4-圈.
英文摘要：
An avoidance problem of configurations in 4-cycle systems is investigated by generalizing the notion of sparseness, which is originally from Erdős' r-sparse conjecture on Steiner triple systems. A 4-cycle system of order v, 4CS(v), is said to be r-sparse if for every integer j satisfying 1 < j < r+1 it contains no configurations consisting of j 4-cycles whose union contains precisely j+3 vertices. If an r-sparse 4CS(v) is also free from copies of a configuration on two 4-cycles sharing a diagonal, called the double-diamond, we say it is strictly r-sparse. In this paper, we show that for every admissible order v there exists a strictly 4-sparse 4CS(v). We also prove that for any positive integer r > 1 and sufficiently large integer v there exists a constant number c such that there exists a strictly r-sparse 4-cycle packing of order v with cv^2 4-cycles.
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