A small embedding for partial even-cycle systems

Let m = 2k. We show that for some 0 ≤ ξ <1, a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (mn)/2 + (2m² 1) √(8n + 1)/4 + 4m^{3 2} + 4 + 1/2. For fixed m, this is asymptotic in n to (mn)/2 and so for large n is roughly one-fourth the best known bound of 2mn + 1. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 205–215, 1999