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

Abstract

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 20.05em … , a n ), allows us to find a solution to O P ( 2 a ∞ − 1 , 2-0.2em a 1 , 2-0.2em a 20.05em … , 2-0.2em a n ) when the a i’s are even ( i ≠ ∞), and O P ( p ( a ∞ − 1 ) + 1 , p-0.2em a 1 , p-0.2em a 20.05em … , p-0.15em a n ) when p is an odd prime, with no restrictions on the a i’s.