Small embeddings of partial directed cycle systems

Abstract

In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.