AI 中文总结
本文针对满足特定Ore型条件的简单图，证明其对任意1≤k≤n-1均存在k-因子，拓展了已有正则因子存在性的相关结论。
AI 中文摘要
设G是阶为n的简单图，满足如下Ore型条件：对G中任意两个不相邻顶点x和y，有d_G(x)+d_G(y)≥n+k-2，其中1≤k≤n-1，kn为偶数，d_G(x)是G中x的度数。已知当k=1或2时，G存在k-因子；Lu和Ning（《图论杂志》，94卷，2020年，307-319页）证明当k≥n/2时，G存在k-因子。本文证明对任意1≤k≤n-1，G都存在k-因子。
英文摘要
Let $G$ be a simple graph of order $n$ satisfying the following Ore-type condition: For any two nonadjacent vertices $x$ and $y$ of $G$, $d_G(x)+d_G(y)\geq n+k-2$, where $1\leq k\leq n-1$, $kn$ is even and $d_G(x)$ is the degree of $x$ in $G$. It is well known that $G$ has a $k$-factor for $k=1$ or $2$. Lu and Ning (J. Graph Theory, 94(2020), 307-319) proved that if $k\geq n/2$, then $G$ has a $k$-factor. In this paper, we show that $G$ has a $k$-factor for any $1\leq k\leq n-1$.
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