On bipartite 2‐factorizations of kn − I and the Oberwolfach problem

It is shown that if F₁, F₂, …, F_t are bipartite 2‐regular graphs of order n and α₁, α₂, …, α_t are positive integers such that α₁ + α₂ + ? + α_t = (n ? 2)/2, α₁≥3 is odd, and α_i is even for i = 2, 3, …, t, then there exists a 2‐factorization of K_n ? I in which there are exactly α_i 2‐factors isomorphic to F_i for i = 1, 2, …, t. This result completes the solution of the Oberwolfach problem for bipartite 2‐factors. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:22‐37, 2011     

1‐rotational k‐factorizations of the complete graph and new solutions to the Oberwolfach problem

We consider k‐factorizations of the complete graph that are 1‐rotational under an assigned group G, namely that admit G as an automorphism group acting sharply transitively on all but one vertex. After proving that the k‐factors of such a factorization are pairwise isomorphic, we focus our attention to the special case of k = 2, a case in which we prove that the involutions of G necessarily form a unique conjugacy class. We completely characterize, in particular, the 2‐factorizations that are 1‐rotational under a dihedral group. Finally, we get infinite new classes of previously unknown solutions to the Oberwolfach problem via some direct and recursive constructions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 87–100, 2008