$\alpha$-可分解4-圈系统

An m-cycle system of order v and index λ, denoted by m-CS（v,λ）, is a collection of cycles of length m whose edges partition the edges of λKv. An m-CS（v,λ） is α-resolvable if its cycles can be partitioned into classes such that each point of the design occurs in precisely α cycles in each class. The necessary conditions for the existence of such a design are m｜λv（v-1）/2,2｜λ（v -1）,m｜αv,α｜λ（v-1）/2. It is shown in this paper that these conditions are also sufficient when m = 4.

$\alpha$-Resolvable Cycle Systems for Cycle Length 4

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 $m=4$.