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 was studied in detail by Buratti and Rinaldi. They found that if admits a 1‐rotational 2‐factorization, then the involutions of are pairwise conjugate. We extend their result by showing that if a finite group admits a 1‐rotational ‐factorization with even and odd, then has at most conjugacy classes containing involutions. Also, we show that if has exactly 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 ‐factorization under given a 1‐rotational 2‐factorization under a finite group . This construction, given a 1‐rotational solution to the Oberwolfach problem , allows us to find a solution to when the ’s are even (), and when is an odd prime, with no restrictions on the ’s.