Solution to the outstanding case of the spouse‐loving variant of the Oberwolfach problem with uniform cycle length

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.