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 2 n ‐factorization under G × Z n given a 1‐rotational 2‐factorization under a finite group G . This construction, given a 1‐rotational solution to the Oberwolfach problem O P ( a ∞ , a 1 , a 2… , a n ) , allows us to find a solution to O P ( 2 a ∞ − 1 ,  2a 1 ,  2a 2… ,  2a n ) when the a i ’s are even ( i ≠ ∞ ), and O P ( p ( a ∞ − 1 ) + 1 ,  pa 1 ,  pa 2… ,  pa n ) when p is an odd prime, with no restrictions on the a i ’s.