The Hamilton-Waterloo problem seeks a resolvable decomposition of the complete graph K n , or the complete graph minus a 1-factor as appropriate, into cycles such that each resolution class contains only cycles of specified sizes. We completely solve the case in which the resolution classes are either all 3-cycles or 4-cycles, with a few possible exceptions when n = 24 and 48. q
A ðK 4 À À eÞ-design on v v þ w points embeds a P 3 -design on v v points if there is a subset of v v points on which the K 4 À À e blocks induce the blocks of a P 3 -design. It is shown that w ! 3 4 ðv v À 1Þ. When equality holds, the embedding design is easily constructed. In this paper, the next case, when w ¼ 3 4 v v, is settled with finitely many exceptions. #
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