马秀文,田子红.$\alpha$-可分解4-圈系统[J].数学研究及应用,2009,29(6):1102~1106
$\alpha$-可分解4-圈系统
$\alpha$-Resolvable Cycle Systems for Cycle Length 4
投稿时间:2007-12-05  修订日期:2008-10-07
DOI:10.3770/j.issn:1000-341X.2009.06.021
中文关键词:    圈系统  $\alpha$-可分解.
英文关键词:cycle  cycle system  $\alpha$-resolvable.
基金项目:国家自然科学基金(No.10971051).
作者单位
马秀文 北京邮电大学网络与交换技术国家重点实验室, 北京 100876 
田子红 河北师范大学数学与信息科学学院, 河北 石家庄 050016 
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中文摘要:
      如果一个完全多重图$\lambda K_{v}$的边集可以分拆为$m$长圈,则称这些圈构成一个$\lambda K_{v}$上的$m$-圈系统,并记作$m$-$CS(v,\lambda)$.若一个$m$-$CS(v,\lambda)$中的$m$长圈可以分拆为若干个$\alpha$-平行类,即$K_{v}$中的每个点在每个类中均恰好出现$\alpha$次,则称该$m$-$CS(v,\lambda)$是$\alpha$-可分解的.$\alpha$-可分解的$m$-$CS(v,\lambda)$存在的必要条件是
英文摘要:
      An $m$-cycle system of order $v$ and index $\lambda$, denoted by $m$-${\rm CS}(v,\lambda)$, is a collection of cycles of length $m$ whose edges partition the edges of $\lambda K_{v}$. An $m$-${\rm CS}(v,\lambda)$ is $\alpha$-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely $\alpha$ cycles in each class. The necessary conditions for the existence of such a design are $m|\frac{\lambda v(v-1)}{2},2|\lambda(v-1),m|\alpha v,\alpha|\frac{\lambda(v-1)}{2}$. It is shown in this paper that these conditions are also sufficient when $m=4$.
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