Abstract: It was proved by S. Win in 1982 that if the sum of the degree of nonadjacent vertices of a simple graph G of order 2n is at least 2n + 1, then G has a Hamilton cycle and a 1-factor which are edge-disjoint. In this paper, it is proved that, under almost the same condition as Win's theorem, G has at least two Hamilton cycles and a 1-factor which are edge-disjoint.