On Monday, September 15, 2003, at 12:05 PM, Thane Plambeck wrote:
I think they're called one factorizations
Yes K_2n always has a one-factorization, with 2n-1 "factors", that's a folk theorem. I'm not sure about cyclic ones
Ah, thank you Thane. Of course -- I knew the definition of k-factorizations, but it didn't occur to me to look things up that way. I now find: --------- Hartman, Alan; Rosa, Alexander Cyclic one-factorization of the complete graph. European J. Combin. 6 (1985), no. 1, 45--48. It is shown that the complete graph $K_n$ has a cyclic 1-factorization if and only if $n$ is even and $n\neq2^t, t\geq3$. Further, the number of cyclic 1-factorizations of $K_n$ for $n\leq16$ is determined. --------- I assume that a "cyclic 1-factorization" is the symmetry I was asking about. Since my first email, I noticed that the method I used for n=6 generalizes to any n=2*odd, but I haven't had the time to think about other cases. That still leaves the counting problem, which I haven't thought about at all. Maybe tomorrow I'll grab a copy of the above paper and see what they did up to n=16. (NJAS fodder, I'm sure...) --Michael Kleber kleber@brandeis.edu