On the generalized Oberwolfach problem

The generalized Oberwolfach problem OP_t(2w + 1; N_1, N_2, …, N_t; α_1, α_2, …, α_t) asks for a factorization of K_{2w + 1} into α_i C_{N_i}-factors (where a C_{N_i}-factor of K_{2w + 1} is a spanning subgraph whose components are cycles of length N_i ≥ 3) for i = 1, 2, …, t. Necessarily, N = lcm(N_1, N_2, …, N_t) is a divisor of 2w + 1 and w = Σ_{i=1}^t α_i.

For t = 1 we have the classic Oberwolfach problem. For t = 2 this is the well-studied Hamilton-Waterloo problem, whereas for t ≥ 3 very little is known.

In this paper, we show, among other things, that the above necessary conditions are sufficient whenever 2w + 1 ≥ (t + 1)N, α_i > 1 for every i ∈ {1, 2, …, t}, and gcd (N_1, N_2, …, N_t) > 1. We also provide sufficient conditions for the solvability of the generalized Oberwolfach problem over an arbitrary graph and, in particular, the complete equipartite graph.