The Hamilton-Waterloo Problem with 4-Cycles and a Single Factor of n-Cycles

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A 2-factor in a graph G is a 2-regular spanning subgraph of G, and a 2-factorization of G is a decomposition of all the edges of G into edge-disjoint 2-factors. A {Cmr, Cns}-factorization of Kυasks for a 2-factorization of Kυ, where r of the 2-factors consists of m-cycles, and s of the 2-factors consists of n-cycles. This is a case of the Hamilton-Waterloo problem with uniform cycle sizes m and n. If υ is even, then it is a decomposition of Kυ- F where a 1-factor F is removed from Kυ. We present necessary and sufficient conditions for the existence of a {C4r, Cn1}-factorization of Kυ- F. © 2012 Springer Japan.

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Graphs and Combinatorics