The intersection problem for m‐cycle systems

Let I m ( v ) denote the set of integers k for which a pair of m ‐cycle systems of K v , exist, on the same vertex set, having k common cycles. Let J m ( v ) = {0, 1, 2,…, t v −2, t v } where t v = v ( v − 1)/2 m . In this article, if 2 mn + x is an admissible order of an m ‐cycle system, we investigate when I m (2 mn + x ) = J m (2 mn + x ), for both m even and m odd. Results include J m (2 mn + 1) = I m (2 mn + 1) for all n > 1 if m is even, and for all n > 2 if n is odd. Moreover, the intersection problem for even cycle systems is completely solved for an equivalence class x (mod 2 m ) once it is solved for the smallest in that equivalence class and for K 2m+1 . For odd cycle systems, results are similar, although generally the two smallest values in each equivalence class need to be solved. We also completely solve the intersection problem for m = 4, 6, 7, 8, and 9. (The cased m = 5 was done by C‐M. K. Fu in 1987.) © 1993 John Wiley & Sons, Inc.