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

Abstract

AbstractA ‐semiframe of type is a ‐group divisible design of type in which is the vertex set, is the group set, and the set of ‐cycles can be written as a disjoint union where is partitioned into parallel classes on and is partitioned into holey parallel classes, each parallel class or holey parallel class being a set of vertex disjoint cycles whose vertex sets partition or for some . In this paper, we almost completely solve the existence of a ‐semiframe of type for all and a ‐semiframe of type for all and .