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Solution to the outstanding case of the spouse‐loving variant of the Oberwolfach problem with uniform cycle length
Journal of Combinatorial Designs ( IF 0.5 ) Pub Date : 2020-11-10 , DOI: 10.1002/jcd.21759
Andiyappan Shanmuga Vadivu 1 , Lakshmanan Panneerselvam 1 , Appu Muthusamy 1
Affiliation  

Let K n + I denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted O P + ( m 1 , m 2 , , m t ) , asks for the existence of a 2‐factorization of K n + I in which each 2‐factor consists of cycles of length m i , for all i , 1 i t , such that n = m 1 + m 2 + + m t . If m 1 = m 2 = = m t = m , then the problem is denoted by O P + ( n ; m ) . In this paper, we construct a solution to O P + ( 4 m ; m ) when m 5 is an odd integer. This completes the proof of the conjecture posed by Bolohan et al. In addition, we find a solution to O P + ( 3 , m ) when m 5 is an odd integer.

中文翻译:

具有均匀周期长度的Oberwolfach问题的热爱配偶变体的杰出案例的解决方案

ķ ñ + 一世 表示重复1因子的偶数阶完整图。Oberwolfach问题的热爱配偶的变体,表示为 Ø P + 1个 2 Ť ,要求存在2因式分解 ķ ñ + 一世 其中每个2因子由长度循环组成 一世 , 对所有人 一世 1个 一世 Ť ,这样 ñ = 1个 + 2 + + Ť 。如果 1个 = 2 = = Ť = ,则问题表示为 Ø P + ñ ; 。在本文中,我们构建了一个解决方案 Ø P + 4 ; 什么时候 5 是一个奇数整数。这就完成了由Bolohan等人提出的猜想的证明。此外,我们找到了解决方案 Ø P + 3 什么时候 5 是一个奇数整数。
更新日期:2020-12-08
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