Neighborhood Conditions and $k$-Factors

Abstract

Let $k$ be an integer such that $k\geq 2$, and let $G$ be a connected graph of order $n$ such that $n\geq 9k-1-4\sqrt{2(k-1)^2+2}$, $kn$ is even, and the minimum degree is at least $k$. We prove that if $|N_G(u)\cup N_G(v)|\geq\frac{1}{2}(n+k-2)$ for each pair of nonadjacent vertices $u$, $v$ of $G$, then $G$ has a $k$-factor.