Resolvable and near-resolvable decompositions ofDK v into oriented 4-cycles

Abstract

A resolvableX-decomposition ofDK v (the complete symmetric digraph onv vertices) is a partition of the arcs ofDK v into isomorphic factors where each factor is a vertex-disjoint union of copies ofX and spans all vertices ofDK v . There are four orientations ofC4 (the 4-cycle), only one of which has been considered: Bennett and Zhang, Aequationes Math.40 (1990), 248–260. We give necessary and sufficient conditions onv for resolvableX-decomposition ofDK v , whereX is any one of the other three orientations ofC4. A near-resolvableX-decomposition ofDK v is as above except that each factor spans all but one vertex ofDK v . Again, one orientation ofC4 has been dealt with by Bennett and Zhang, and we provide necessary and sufficient conditions onv for the remaining three cases. The construction techniques used are both direct (for small values ofv) and recursive.