Nesting of cycle systems of odd length (Q750445)

A nesting of an m-cycle system \((K_ n,L)\) is equivalent to an edge decomposition of \(2K_ n\) into wheels, each with m spokes with the property that for each pair of vertices x and y, one of the edges \(\{\) x,y\(\}\) occurs on the rim of a wheel and one of the edges \(\{\) x,y\(\}\) is a spoke of a wheel. In this paper the authors prove that for any odd m, the spectrum of m-cycle systems which can be nested is the set of all \(m\equiv 1\) (mod 2m) with at most 13 possible exceptions for each m. Some of these exceptions are then removed for some small values of m. For a recent survey of nested cycle systems, see the third author [The construction of nested cycle systems, Coding Theory and Design Theory, Part I: Coding theory, Proc. Workshop IMA Program Appl. Comb., Minneapolis/MN (USA) 1987-88, IMA Vol. Math. Appl. 20, 362-367 (1990)].