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- W4289367688 abstract "The concept of a $1$-rotational factorization of a complete graph under a finite group $G$ was studied in detail by Buratti and Rinaldi. They found that if $G$ admits a $1$-rotational $2$-factorization, then the involutions of $G$ are pairwise conjugate. We extend their result by showing that if a finite group $G$ admits a $1$-rotational $k=2^nm$-factorization where $ngeq 1$, and $m$ is odd, then $G$ has at most $m(2^n-1)$ conjugacy classes containing involutions. Also, we show that if $G$ has exactly $m(2^n-1)$ conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a $1$-rotational $2n$-factorization under $G times mathbb{Z}_n$ given a $1$-rotational $2$-factorization under a finite group $G$. This construction, given a $1$-rotational solution to the Oberwolfach problem $OP(a_{infty},a_1, a_2 cdots, a_n)$, allows us to find a solution to $OP(2a_{infty}-1,^2a_1, ^2a_2cdots, ^2a_n)$ when the $a_i$'s are even ($i neq infty$), and $OP(p(a_{infty}-1)+1, ^pa_1, ^pa_2 cdots, ^pa_n)$ when $p$ is an odd prime, with no restrictions on the $a_i$'s." @default.
- W4289367688 created "2022-08-02" @default.
- W4289367688 creator A5022955004 @default.
- W4289367688 creator A5065757718 @default.
- W4289367688 date "2018-10-23" @default.
- W4289367688 modified "2023-09-24" @default.
- W4289367688 title "A Method to Construct $1$-Rotational Factorizations of Complete Graphs and Solutions to the Oberwolfach Problem" @default.
- W4289367688 doi "https://doi.org/10.48550/arxiv.1810.10047" @default.
- W4289367688 hasPublicationYear "2018" @default.
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