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On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

Given two 2‐regular graphs F1 and F2, both of order n, the Hamilton‐Waterloo Problem for F1 and F2 asks for a factorization of the complete graph Kn into α1 copies of F1, α2 copies of F2, and a 1‐factor if n is even, for all nonnegative integers α1 and α2 satisfying α1+α2=⌊n−12⌋ . We settle the Hamilton‐Waterloo Problem for all bipartite 2‐regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2‐regular graph of the same order.