Win [\emph{J. Graph Theory} {\bf 6}(1982), 489--492] conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\geq k+2$. In this paper, we prove that Win's conjecture is true for $k\geq n/2$, where $n$ is sufficiently large. To show this result, we prove a theorem on $k$-factor in a graph under some Ore-type condition. Our main tools include Tutte's $k$-factor theorem, the Karush-Kuhn-Tucker theorem on convex optimization, and the solution to the longstanding 1-factor decomposition conjecture.

中文翻译：

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大 k 因子和不相交完美匹配的矿石类型条件

赢 [\emph{J. 图论} {\bf 6}(1982), 489--492] 推测 $n$ 个顶点上的图 $G$ 包含 $k$ 不相交的完美匹配，如果任何两个不相邻顶点的度和至少为 $ n+k-2$，其中 $n$ 是偶数，$n\geq k+2$。在本文中，我们证明 Win 猜想对于 $k\geq n/2$ 是正确的，其中 $n$ 足够大。为了证明这个结果，我们证明了在某种矿石类型条件下的图中$k$-factor 的一个定理。我们的主要工具包括 Tutte 的 $k$-factor 定理、关于凸优化的 Karush-Kuhn-Tucker 定理以及长期存在的 1-factor 分解猜想的解决方案。