On the Hamilton-Waterloo Problem with triangle factors and C₃ₓ-factors

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The Hamilton-Waterloo Problem (HWP) in the case of Cₘ-factors and Cₙ-factors asks if Kᵥ, where v is odd (or Kᵥ-F, where F is a 1-factor and v is even), can be decomposed into r copies of a 2-factor made either entirely of m-cycles and s copies of a 2-factor made entirely of n-cycles. In this paper, we give some general constructions for such decompositions and apply them to the case where m=3 and n=3x. We settle the problem for odd v, except for a finite number of x values. When v is even, we make significant progress on the problem, although open cases are left. In particular, the difficult case of v even and s=1 is left open for many situations.