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

Let Kn+I denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted OP+(m1,m2,…,mt), asks for the existence of a 2‐factorization of Kn+I in which each 2‐factor consists of cycles of length mi, for all i,1≤i≤t, such that n=m1+m2+⋯+mt. If m1=m2=⋯=mt=m, then the problem is denoted by OP+(n;m). In this paper, we construct a solution to OP+(4m;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 OP+(3,m) when m≥5 is an odd integer.