A method to construct 1-rotational factorizations of complete graphs and solutions to the oberwolfach problem

The concept of a 1-rotational factorization of a complete graph under a finite group G was studied in detail by Buratti and Rinaldi. They found that if G admits a 1-rotational 2-factorization, then the involutions of G are pairwise conjugate. We extend their result by showing that if a finite group G admits a 1-rotational k-factorization with k = 2(n)m even and m odd, then G has at most m (2(n) - 1) conjugacy classes containing involutions. Also, we show that if G has exactly m(2(n) - 1) conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1-rotational 2n-factorization under G x Z(n) given a 1-rotational 2-factorization under a finite group G. This construction, given a 1-rotational solution to the Oberwolfach problem OP(a a(1), a(2) ...a(n)), allows us to find a solution to OP(a(infinity), a(2), ..., a(n)) when the ai's are even (i not equal infinity), and OP(p(a(infinity) - 1) + 1, (P)a(1), (P)a(2)..., (P)a(n)) when p is an odd prime, with no restrictions on the ai's.