On the existence of k $k$ -cycle semiframes for even k $k$

Abstract

A $C_k$-semiframe of type $g^u$ is a $C_k$-group divisible design of type $g^u\left(X,G,ℬ\right)$ in which $X$ is the vertex set, $G$ is the group set, and the set $ℬ$ of $k$-cycles can be written as a disjoint union $ℬ=P\cup Q$ where $P$ is partitioned into parallel classes on $X$ and $Q$ is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition $X$ or $X⧹G_j$ for some $G_j\in G$. In this paper, we almost completely solve the existence of a $C_{4k}$-semiframe of type $g^u$ for all $k\ge 1$ and a $C_{4k+2}$-semiframe of type $g^u$ for all $k\ge 1$ and $g\equiv 0\left(\mathrm{mod}8k+4\right)$.