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

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Abstract

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 \({\{C_{m}^{r}, C_{n}^{s}\}}\) -factorization of K υ asks for a 2-factorization of K υ , 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 K υ F where a 1-factor F is removed from K υ . We present necessary and sufficient conditions for the existence of a \({\{C_{4}^{r}, C_{n}^{1}\}}\) -factorization of K υ F.

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Correspondence to Sibel Özkan.

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Keranen, M.S., Özkan, S. The Hamilton-Waterloo Problem with 4-Cycles and a Single Factor of n-Cycles. Graphs and Combinatorics 29, 1827–1837 (2013). https://doi.org/10.1007/s00373-012-1231-6

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  • DOI: https://doi.org/10.1007/s00373-012-1231-6

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