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The intersection problem for m-cycle systems

✍ Scribed by Elizabeth J. Billington

Publisher: John Wiley and Sons
Year: 1993
Tongue: English
Weight: 794 KB
Volume: 1
Category: Article
ISSN: 1063-8539
DOI: 10.1002/jcd.3180010605

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✦ Synopsis

Let Z,(v) denote the set of integers k for which a pair of m-cycle systems of K , exist, on the same vertex set, having k common cycles. Let J,(v) = { 0,1,2, . . . ,t, -2, t,} where t , = v(vl ) / 2 m . In this article, if 2mn + x is an admissible order of an m-cycle system, we investigate when Zm(2mn + x) = Jm(2mn + x), for both m even and m odd. Results include Jm(2mn + 1) = Zm(2mn + 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 Zm) once it is solved for the smallest in that equivalence class and for 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.)

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