On the spouse-loving variant of the Oberwolfach problem

摘要

We prove that Kn+I, the complete graph of even order with a 1-factor duplicated, admits a decomposition into 2-factors, each a disjoint union of cycles of length m = 5 if and only if m divide n, except possibly when m is odd and n=4m. In addition, we show that Kn+I admits a decomposition into 2-factors, each a disjoint union of cycles of lengths m1, horizontal ellipsis ,mt, whenever m1, horizontal ellipsis ,mt are all even.