Cycle systems without 2‐colorings

In an m-cycle system C of order n the blocks are the vertex sets of n(n−1)/(2m) cycles Ci such that each edge of the complete graph Kn belongs to precisely one cycle Ci E C. We prove the existence of m-cycle systems that admit no vertex partition into two classes in such a way that each class meets every cycle of C. The proofs apply both constructive and probabilistic methods, and also some old and new facts about Steiner Triple Systems without large independent sets. © 1996 John Wiley & Sons, Inc.