Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n 1 k, n is odd, and 2 k divides n(n -1). We show that if these necessary conditions are sufficient for all n satisfying k I n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n = 1, 15, 21, or 25 (mod 301, and there exists a 21-cycle system of order n if and only if n = 1, 7, 15, or 21 (mod 421, n.# 7, 15.

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In this paper w e use the Hilton method of amalgamations to give a different proof of a theorem of Nash-Williams that finds necessary and sufficient conditions for the embedding of an edge-colored K, into an edge-colored Kn in which the edges of each color induce a 2-edge-connected k-factor. 0 1995 John Wiley & Sons, Inc.

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